Despite increasing criticism of President Reagan's even-handed approach to the British-Argentine dispute over the Falkland Islands, U.S. officials insist that it offers the best hope of averting an explosion that could do incalculable harm to the interests of all three countries.

Since Monday, when Reagan said, "We're friends of both sides," he has been subjected to a torrent of press criticism questioning the wisdom and propriety of U.S. failure to speak out more in support of Britain, which has been the administration's staunchest ally in Europe.

The columnists, editorial writers and cartoonists taking shots at the president have found considerable additional ammunition in the fact that, in contrast to democratic Britain, Argentina is controlled by a military dictatorship that has been widely denounced for persistent human rights violations and that sought to settle the Falklands issue by occupying the islands through force.

In the face of these arguments, the U.S. strategy, which currently centers on Secretary of State Alexander M. Haig Jr.'s effort to carve out an "honest broker" role between London and Buenos Aires, is described by one official as an attempt to be "quiet and effective rather than moralistic and ineffective."

In short, the administration is working on the assumption that, by avoiding the impulse to point fingers and preach, it may be able to defuse the crisis before it reaches the point where a clash is inevitable and the United States will be forced to choose sides between two countries that it considers important to American policy goals.

Having to make that choice could still prove unavoidable. But, U.S. officials are quick to point out that, for the moment at least, both Britain and Argentina, which view the United States as the only country in a position to play a successful mediating role, have tacitly endorsed Reagan's efforts to avoid tilting overtly toward one side or the other.

The officials said that continues to be the case despite British annoyance at such questionable actions as the attendance by U.N. Ambassador Jeane J. Kirkpatrick, Deputy Secretary of State Walter J. Stoessel Jr. and other senior administration officials at an Argentine Embassy dinner on the day that Argentina invaded the Falklands.

In an interview yesterday with United Press International, British Ambassador Sir Nicholas Henderson said: "I wouldn't have done so. If I had been asked by the Iranian Embassy to go to a banquet the night your hostages were taken, I wouldn't have done so."

In fact, many of the questions raised about the propriety of Reagan's stance appear to be rooted in the continuing opposition of American liberals to the administration's controversial policy of cultivating close ties with Argentina as a counter to alleged communist penetration of the hemisphere. To these critics, the Falklands incident is an additional reason for dropping that approach and keeping Argentina at arms' length.

That point has been debated within the administration, where officials concerned primarily with the strength of the Atlantic alliance have questioned whether U.S.-British ties should be subordinated to efforts to win Argentine support for the administration's Central America policy.

These officials are concerned that the Falklands crisis could cause the fall of Prime Minister Margaret Thatcher's government and its replacement by one that would oppose the top-priority U.S. emphasis on a tough stance toward the Soviet Union and the deployment of American medium-range nuclear missiles in Western Europe.

Administration sources, while stressing that no decisions have been made, also acknowledge privately that if a confrontation cannot be avoided between Britain and Argentina, the overwhelming importance of these considerations to U.S. policy, plus taking into account justice and traditional ties, almost certainly would force Washington to come down on Britain's side.

But, senior administration policymakers, who also attach great weight to the importance of turning back what they perceive as a threat to the Western Hemisphere, also are reluctant to lose Argentina's support and possibly antagonize the rest of Latin America.

For that reason, the administration has taken the line that it still has time before it is forced to make such an agonizing choice and that, in the interim, it could do the greatest service to everyone concerned by retaining the confidence of both countries and using it to seek a peaceful resolution.

Whether that is possible won't become clear until Haig completes his soundings in the two capitals. In the meantime, though, Britain has refrained from calling on the United States for a stronger show of support, and Argentina has not made good on its threat to invoke an inter-American treaty that it contends would compel the United States to come to its aid.

That, U.S. officials contend, is a clear sign that both countries think Reagan is taking the right course and want him to continue on it as long as there is any hope that the United States can find the formula for steering them away from a collision.

In fact, some officials asserted last night, that appears to be a view shared by many other countries with a keen interest in the dispute, including the 30 member-nations of the Organization of American States. The officials noted that the OAS yesterday postponed until Monday a special meeting on the Falklands dispute in order to give Haig more time to carry out his mission.