Tourism in the Bahamas, a multi-island nation that is so close as 60 miles to Florida's eastern shore yet lies scattered across some 100,000 square mile of the North Atlantic, has been feeling the effects worldwide economic problems.

Nassau, the capital, on New Providence Island, reports an estimated 15- to 20-percent fall-off in visitors for the first 10 months of 1981 compared to 1980, which was a good year. Freeport, on Grand Bahama, apparently fared worse; some hotel operators say tourism for the same period is down between 30 and 45 percent from last year, though government figures show less of a decline. The "Out Islands," now officially known as the "Family Islands," are doing better due to stops by cruise vessels. (The new name covers the 700 or so islands and cays other than New Providence and Grand Bahama that comprise the Bahamian archipelago.)

Obviously, the welcome mat has been dusted off with a flourish as government and private industry work even harder to sell their theme: "It's better in the Bahamas."

Travelers from the United States still make up 80 percent of all visitors to the Bahamas, though their numbers from the Northeast and Midwest have dropped because of increased air fares. Most Canadians, normally 10 percent of the total, have now stopped coming to a number of formerly favored vacation spots, and European traffic to the islands has decreased from more than 10 percent to between 5 and 10 percent of the total -- a pronounced jolt, since visitors from Europe will vacation for from two to six weeks while North Americans tend, on the average, to limit their stay to only three to seven days.

The culprits are largely the familiar budgetary ones common to many countries and individuals today: inflation, unemployment, uncertainty, and the rising cost of travel. The discontinuation of direct flights from Europe to Nassau also has made trip planning abroad more cumbersome, particularly when time is an important factor.

But direct competition from the Bahamas' best customer must also be reckoned with, notes Colin Tatem, director of public relations and marketing for the posh Princess Hotels, in Freeport.

"The U.S. itself is going into tourism," Tatem observes, "and now not only has large numbers of Americans traveling within its own borders, but also is attracting large groups of South Americans and Europeans."

Nonetheless, I found a basic core of optimism during a November visit to the islands. Unless the U.S. winter proves unusually warm, or a more severe economic downturn intervenes, the Bahamas look for an improved -- though not dramatically so -- 1981-1982 season.

It's important to point out that the Bahamas lie in the Atlantic, not the Caribbean. Thus, while the Gulf Stream and southerly latitudes usually insure that air and water temperatures are very pleasant during the months most of the United States is shivering, sometimes New Providence and Grand Bahama islands can be affected when a cold front strikes Miami.

If optimism can be measured in steel and concrete, there's enough to carry over into any number of seasons to come. Heavy foreign investment -- American, Canadian and European for the most part -- is helping realize Bahamian plans for growth. In the Nassau area alone, new construction expenditures are in the hundreds of millions of dollars.

For example, the prestigious, 250-room Britannia Beach Hotel on Paradise Island just north of Nassau (daily winter rates: single or double, $140), is adding 352 rooms and several new restaurants and bars scheduled for completion this winter at a cost of $18 million. The 730-room Cable Beach Hotel is expected to open its doors in 1983. A government-owned casino is also in the works; plans include a 1,000-seat theater. And two 18th-century landmarks, Fort Fincastle and Fort Charlotte, are getting long-needed facelifts.

Freeport exudes the same confidence in the face of economic woes. "We're probably hardest hit, down 30 percent for the whole year," confides Anton Gontsche, general manager of the Moorish-style Princess Tower Hotel (daily winter rates: single or twin, $74-$109). Despite hard times, the Princess Tower has already spent $2.5 million on a complete renovation and several new restaurants for the 400-room luxury hotel.

Behind all this frenetic planning and building are a few positive notes contributing to a general feeling of "It's got to get better."

Sea arrivals -- admittedly of greater economic import for shops and restaurants than for hotels -- were up a reported 10 percent for the first eight months of 1981. In January 1982, Scandinavian World Cruises is scheduled to initiate a cruise/car ferry service between Miami and Freeport; a similar service is due later in the year between New York and Freeport. Also starting in January, the Norwegian Caribbean Line's S.S. Norway will call weekly at Nassau on its regular itinerary from Miami.

While the economic climate remains a question mark, an increase in the number of airline seats, despite the effects of the air traffic controllers' strike, would seem to make hotel expansions a less-risky proposition.

Eastern Airlines, the major carrier into the Bahamas, has switched to larger-capacity aircraft and now offers more seats than before the strike, according to Bahamian officials. In addition, new air service will be available beginning this month. Delta has scheduled flights from Boston to Newark, and then nonstop to Nassau, beginning Dec. 1, and will also begin service to Nassau from Detroit via Atlanta on that date. American Airlines will fly nonstop from Dallas to Nassau beginning Dec. 17, and Air Florida will inaugurate nonstop service to Freeport from Newark on Dec. 1.

In these days of rising costs, it seems only fair to ask: Is a vacation in the Bahamas a "good buy"?

"We're not a high-priced destination," maintains Gontsche of the Princess Tower. "Our rates are almost 50 percent lower than those in Nassau." Nassau, being the capital, is naturally the center of political and social activity and therefore the rates of many of its hotels are higher.

Freeport, an American creation dubbed "The Riviera of the Bahamas," is essentially a giant resort complex about a quarter of a century in the making. It boasts more than 60 tennis courts and 126 holes of championship golf. Some of the world's finest reefs for diving are here, explored yearly by nearly 12,000 diving enthusiasts. The UNEXSO Diving School offers a resort scuba-diving course spread over two days for $49, highlighted at the end by a guided visit to a reef to try out one's newly acquired skills.

Freeport/Lucaya covers about 200 square miles or half the total area of the island. It was conceived by Wallace Groves, a New York stock market operator, who persuaded the British government to issue him a Crown Charter so he could form the Grand Bahama Port Authority, Ltd. After gaining independence, the Bahamian government continued most of the special privileges that make the area a hybrid, semi-autonomous appendage to the rest of the island, including freedom from all taxes and all import duties except on items for personal consumption. Benquet, an international combine based in the Philippines, now owns a majority interest in the Authority.

Freeport is famed among shoppers for its Moorish-style International Bazaar -- a medley of 70 shops and restaurants combining the wares and cuisines of their distant counterparts in Europe, Latin America, Africa and the Far East. In Freeport, too, are the Gardens of the Groves, with a display of 10,000 species of plants and shrubs, and the new Grand Bahama Museum.

It is necessary to shop wisely (this is not a duty-free port). Before leaving home, compare discount prices for specific items like cameras and watches if you want to be certain that a purchase here is really a "bargain" these days.

Clearly, a "good buy" must be seen not only in terms of purchases but also with an eye on the fringe benefits -- the recreational and cultural opportunities a location provides. Nassau and Freeport offer a wide variety of both.

While at Freeport I was reminded of what, for many travelers to the Bahamas, including myself, has proved to be a bonus feature of their vacation. I stopped by the Ministry of Tourism and chatted with Terrecita Kemp, head of the local People-to-People Programme -- an informal means of introducing visitors to island people sharing similar interests. She arranged for me to meet Maquella Smith, a cultural affairs officer with the Ministry of Education and Culture.

We enjoyed a delightful hour discussing her work in planning international art exhibits and leading art seminars for young people. This past summer in Nassau, under the same program, I had spent an afternoon sampling Bahamian foods with Cindy Williams, deputy program director of radio station ZNS, and her husband Mackey, an engineer by profession, composer of native music, and playwright. There followed an evening with internationally famed De'Ynza Burrows, better known as "King Conch," and his family, who introduced me to the many ways of cooking conch, a Bahamian delicacy.

The People-to-People Programme, now available only in Nassau (2-7500-6) and Freeport (352-8044), handles arrangements without charge. Application for visits can be made prior to leaving home through the Bahamas tourist office in Washington (659-9135), through social directors of cruise lines serving the islands, or in the Bahamas by phoning or visiting local tourist offices or through social hostesses at hotels. Visits can often be arranged within two days.

A good buy at either destination is still possible today in terms of total vacation costs. Special charter packages, designed by travel agents, airlines, and tour operators to encourage travel to the Bahamas, provide bargain rates at selected hotels.