When British Foreign Secretary Lord Carrington toured Latin America this summer with a group of high-powered British businessmen, he sandwiched in a special overnight stop in this former colony between longer visits to bigger and more important nations such as Brazil and Venezuela.

Carrington's trip here followed a visit by Barbadian Prime Minister Tom Adams to President Carter last April, a meeting Washington finally agreed to after two years of requests by U.S. Embassy and local officials in Bridgetown.

There was no ostensible reason for either visit; no trouble for Carrington to look into here and little private business to be done. Carter, in fact, had other things on his mind when Adams showed up. That day, Carter announced the aborted U.S. rescue mission in Iran.

Barbados, in contrast to some of its Caribbean neighbors, has enjoyed an independence of stability, growth and strong friendship with both the former ruler and the hemisphere's dominant power. The sudden surge of attention to a small, relatively inconsequential state indicates their growing realization that such friendship and stability can no longer be taken for granted in this part of the world.

Tied by a common history of slavery and colonialism, the islands of the Caribbean now are bound in a brotherhood of economic difficulties and, for many, abject poverty. All are needy, and many are among the newest of the world's nations.

Cut adrift from the European colonial powers that provided for their defense, made their decisions, bought their products and kept their books, they are faced with the equally unattractive alternatives of finding their way in a power vacuum, or selling themselves to the highest big power bidder.

Caribbean leaders complain that the only time the "big countries" pay attention to them is during periods of heightened concern about Cuba, and the current preoccupation is no exception. There is a belief that Fidel Castro has embarked on a new campaign of "adventurism" to bring the cold war to his tropical basin whose dozens of islands -- with the notable exception of his own -- have been considered the sole preserve of the West.

Viewed from Washington, the possible consequences of large-scale expansion of Cuban -- and by extension Soviet -- influence in the Caribbean are catastrophic.

The Caribbean islands lie scattered across the approaches to the Panama Canal and the shipping lanes through which more than half the United State's imported oil passes.

From Cuba's western tip, a short hop from Mexico's Yucatan Peninsula and the Florida Keys, the islands stretch to Trinidad, a ferry ride across a narrow channel to the South American mainland in Venezuela. Their territories offer a multitude of opportunities for the establishment of unfriendly ideological beachheads and military bases.

Some of the islands already have found elements of Castro's model attractive, Jamaica, under Prime Minister Michael Manley, is a "socialist democracy" whose government structure and financial aid still come from the West, but which sees Cuba as a friend and Third World mentor.

Even more starting for the West was the English-speaker Caribbean's first coup, in March 1979. The new leftist government of the tiny spice island of Grenada, itself a British dependency until 1974, is outspokenly hostile to the United States, and its People's Revolutionary Army vows to fight against "imperialism."

Such an event would have been unthinkable before the victory of the Cuban revolution in 1959. It had seldom seriously occurred to anyone since.

But much has changed in this sleepy sea in the past 20 years, not the least of which is the fact that in 1959 only three Caribbean islands were independent. Today, there are 11. Three of them -- St. Vincent, St. Lucia and Dominica -- gained independence in the past wo years alone. More new states are still to come as Britain, which retains four outright colonies in the Caribbean and several islands in varying degrees of dependency, further retreats from empire.

The rash of independence has upset a careful balance of power in the Carribbean, where the United States traditionally took responsibility for the large northern islands -- including military occupations early this century of Cuba, Haiti and the Dominican Republic -- while the British kept their smaller sized but more numerous flock in line farther to the south.

The "loss" of Cuba from this equation initially was upsetting, but as the years passed and Cuban attention began to be diverted elsewhere, the worry that Cuba would be emulated by the other islands largely passed. Fears of a communits-inpsired takeover in the Dominican Republic were allayed with the intervention of U.S. troops in 1965.

Although all but a few British colonies became "associated states" in 1967, giving them internal self-government on the road to independence, Britain retained responsibility for their defense and foreign relations. Today, most of them are on their own in all respects.

As tension and concern over the Caribbean have increased, the British and Americans have tended to grow cranky with each other.

"We see now that the British are eventually going to drop the ball," in their former Caribbean colonies, a Carter administration official said in Washington. "They are not exercising the authority we had hoped; they feel like they want to cut and run and get out as fast as they can."

There are vague U.S. rumbles that the British do not want to "spend the money" it would take to keep the islands stable.

These kinds of charges exasperate the British.

"The point of all this is that these islands and governments ask for independence, and if they think we delay, they are very bitter," said a regional British official. "In the face of United Nations decisions" on decolonization, "what are we supposed to do?" he asked.

With all due respect, it was the United States that kept urging us to get out," as the idea of modern states retaining colonies became unfashionable, he said.

Whatever the reasons, the faults or the desirability of the birth of the Caribbean's new nations, there is agreement that they must be dealt with both for their own sakes and U.S. security. New and old, their shores are being lapped by an undercurrent of change, of which Grenada and Jamaica are but the earliest and most visible results.

The Caribbean islands' colonial legacy is one of a cacophony of languages -- Spanish, English, French, Dutch -- a collection of island patois tongues that are endless variations of each. Their systems of government range from Cuba's totalitarian communist state to Barbados' British parliamentarianism to U.S. style representative democracy in the Dominican Republic and dynastic autocracy in Haiti.

What they all have in common is the newness of their civilization. All are populated by former African slaves brought to farm the large colonial plantations. Most inhabitants eveutually interbred to varying degrees with the Europeans who colonized the islands and in the case of some of the British territories, with Asians who were themselves colonial subjects.

While some, like Barbados, have managed their economies better than others, all are poor. Unemployment, particularly among the young, runs as high as 40 percent onsome of the islands. Birth rates are high; housing is poor.

At the same time that the prices of their agricultural exports have remained fairly stationary, the cost of imported oil has become the bane of their existences and threatens to wipe out small economic gains made by some in the early 1970s.

A recent report by the Caribbean Group, a U.S.- and World Bank-backed organization of aid donor and recipient nations, along with international and regional banks, gives little cause for encouragement.

Starting from the southernmost islands, it notes that "the growth rate . . . in Grenada is estimated at about 2 percent in 1979, compared with 5 percent the previous year." The report adds:

Farther north, "economic activity [in Haiti] in 1979 suffered the combined effect of poor weather and of production cycle." In Jamaica, "the the downswing of the two-year coffee performance of [the] economy has been, on the whole, disappointing."

The dry, economic language of the report pinpoints the realities of fiscal life in the Caribbean -- single crops whose failures lead to disaster, hurricanes, volcanoes, "political instability . . . decline in confidence . . . weaknesses in public finance and public administration." There are few bright spots and no indications that economic hardship is significantly affected by a country's system of government.

By most reckoning, the island states of the Carribbean, particularly the former British colonies in the southereastern area, are unviable units, little more than historical afterthoughts and appendages to the international economic system.

As one Jamaican politician pointed out, "20 of thse places could fit inside the King Ranch in Texas."

Both the United States and Cuba are well aware that such problems and attitudes create "opportunities" for outside interference.

The question for the West has been whether to first address the problems themselves, eliminating the opportunities through economic aid and development, or to come down hard and heavy on the presumed opportunists by increasing Western intelligence and military presence in the region and helping sympathetic local governments to increase theirs.

Early in 1977, the then-new Carter administration named a special task force to study Caribbean problems. One of the priorities it set was a sharp increase in economic aid to the region, and the level of U.S. assistance rose accordingly from $70 million in 1976 to $130 million in 1979.

But if what has happened in Jamaica and Grenada is considered evidence that Castro is bringing the cold war to the Caribbean, the United States has responded in kind.

On Oct. 1, following the administration's "discovery" of a division of Soviet combat troops in Cuba, Carter announced U.S. action to "resolve these problems and counter these activities," including an expansion of the U.S. military presence in the area and an "increase [in] our economic assistance to alleviate the unmet economic and human needs in the Caribbean region and further to ensure the ability of troubled peoples to resist local turmoil and possible communist domination."

The realization of that pledge, due to what one administration official called "competing priorities" in the government, has been somewhat lopsided. The administration has requested $149.6 million in economic assistance for the Caribbean in fiscal 1981 -- after inflation, a bare-bones increase over the 1980 total of $136.7 million. More than half the new request is programmed for bilateral assistance for Haiti and the Dominican Republic, both already considered firmly in the U.S. camp.

At the same time, requested military assistance has jumped from $1.4 million actually approved in 1979 to a requested $9.1 million for fisacl 1981.It includes $5 million for equipment in a first U.S. military aid program in Barbados.

Some U.S. officials in Washington have been encouraged by what they see as a "conservative sweep" in the southeastern islands in four general elections held over the past 10 months Conservative or centrist governments in St. Vincent, St. Kitts-Nevis, Antigua and Dominica all won wide margins over leftist parties, whose ideology is modeled after Cuba's in most cases and in some case whose funding is believed to come in part from Cuban President Fidel Castro.

Here under the bright sun, however, the results of those elections are looked at somewhat differently, almost as a given in part of the world where politics reflect little of the growing pressures and undercurrents of change.

The comments, from varied souces, are all strikingly similar.

"I wouldn't say I'm worried about the Cuban-oriented Marxists" in the islands, said a British diplomat. "They're not gaining any ground through voting; they have no more chances of taking over here than the U.S. Communist Party does there. And nobody thinks the Cubans are going to do it by 'moving in' militarily.

"The worry is how much longer a growing population can maintain a standard of living with an agricultural base of income and the high aspirations of the young."

The "conservative sweep," said a U.S. diplomat stationed in the region, is "what [the State Department] wants to believe. But the broad middle ground in the Caribbean that is pro-Britain, pro-United States and Canada is going to be eaten away as the economic situation continues to worsen. All the conditions that cause instability around here are going to worsen.

Island officials note that, unlike in Central America, where entrenched rightist military and economic elites are currently being battled by the left, the Caribbean by and large is a region of existing democratic governments that are near the point of exhausting their options in dealing with economic problems. Grenada, they note, was an exception to this rule not because it is not poor but because it was ruled by a despot who showed no inclination to give up power by any means other than by force.

"We don't believe there is any such thing in the Caribbean as a conservative sweep," said a high-level Barbadian official. "Most of the parties are already strongly worker oriented, and the chances of a leftist ideology succeeding . . . are not very bright if people are given a choice.

"Sometimes we don't think the United States understands this trend in the region very deeply," he said. "We really aren't much interested in the rhetoric of the cold war. The basic problems are economic and social."

In presenting alternatives and possible ways out, he said, "I suppose the Cubans are doing exactly what countries do who are trying to win friends and influence people."