Washington — State Department officials think that the damage done by the Falklands crisis to US relations with Latin American countries can be largely overcome if the crisis ends shortly.
But should that conflict drag on, with further loss of life, the damage might be considerably increased, they say. Some experts outside the government, meanwhile, say that the damage caused by the US tilt in favor of Britain has already been severe.
Almost all the experts seem to agree that the damage done to relations with Argentina could last for years and that it has opened up opportunities there for the Soviet Union. There also seems to be agreement that an old friend of the US, Venezuela, is at the top of the list of those who have been offended by American support for Britain. A State Department official said that Peru's reaction against the US presented equally difficult problems for Washington.
The most serious fallout from the Falklands so far has been the openings which it has given the Soviets, say some of the experts. Several of them predicted that the Soviets will soon be in the business of selling weapons to Argentina.
A State Department official acknowledged that the Soviets can offer modern weapons to Latin American nations at prices and on credit terms which the US and other Western nations cannot match. But he said that much still depended on finding a solution to the Falklands crisis which would not result in the humiliation of Argentina. If that nation felt totally isolated from the West, the official said, then, of course, the Soviets might be able to move in with weapons sales.
The State Department official suggested that what the US must do to prevent this kind of reaction is to keep a constant dialogue going with the Latin American nations, including Argentina. Another official said that the US was still ''very much involved'' in helping to find a solution to the Falklands crisis. Secretary of State Alexander M. Haig Jr. was reported to have sent a secret envoy to Argentina only a few days ago to confer with Argentina's leaders. The envoy, Gen. Vernon Walters, a Spanish-speaking ambassador-at-large at the State Department, has undertaken other such secret missions on several occasions in the past.
William D. Rogers, a Washington lawyer and former undersecretary of state, thinks the key to limiting the damage in Latin America is for the US to persuade Britain to state that it has no pretensions to permanent sovereignty over the Falklands. At the same time, however, he asserts that whatever the outcome in the Falklands, Washington's leadership among the Latin American nations has been diminished.
Jack Child, assistant dean at American University's School of International Service, suggests that the damage has already been severe. He says he believes that Venezuela will now be less anxious to support President Reagan's Caribbean Basin economic initiative and that any future American attempts to call for collective action with the Latin Americans under the Rio Treaty of 1947 will be rejected unless it is clearly in the Latins' self-interest.
''The next time the US tries to invoke the Rio Treaty, it will be laughed out of the room,'' said Child.
At the same time, however, he said that much depended on how the Falklands crisis was resolved and on how much military support the US gave to Britain. He also said that while most Latin American nations have at the level of public rhetoric strongly supported Argentina, their private reaction has been decidedly mixed. Many resent the way in which Argentina went about taking the Falklands.
State Department officials take a calm view. They argue, for instance, that Venezuela has shown no signs of faltering in its support for the Caribbean Basin initiative. They assert that the concept of collective action against outside threats to the region remains alive and well. They also correctly contend that reaction among a number of key Latin American nations to the Falklands crisis has hardly been monolithic:
* Mexico has been measured and cautious in its reaction.
* Colombia has strongly supported Secretary Haig's mediation efforts.
* Peru, according to a State Department official, put forth proposals to resolve the crisis which were ''constructive and even-handed.''
* Chile, behind an officially lukewarm attitude, is actually hostile to Argentina. Some Chilean diplomats privately confess that they would like to see Argentina get ''cut down to size'' by the Falklands crisis.
* Brazil, the biggest of the South American nations, while formally siding with Argentina, has acknowledged that both Britain and Argentina have ''honorable and just requirements.'' That position, agreed to by Brazilian President Joao Figueiredo during his talks with President Reagan on May 12, pleased US officials.
* Venezuela is seen as a problem, partly because of its claims to former British territory now held by Guyana and partly because it considers itself a leader among Latin American nations obliged to combat anything smacking of a return to colonialism.