Despite an appeal by President Reagan yesterday that hemispheric nations not let their sympathies for Argentina harm their good relations with the United States, a majority of Latin American countries demanded early today that the United States stop aiding Britain in the Falkland Islands crisis.

The 17-to-0 vote followed hours of debate at a special meeting here of the Organization of American States. At issue was a resolution demanding that the United Kingdom cease immediately its "unjustifiable attacks" and that the United States lift its economic and political sanctions against Argentina.

Reagan, speaking to a U.S.-Mexico interparliamentary conference in Santa Barbara, Calif., tried to calm the rift that the Falklands crisis is causing in U.S relations with Latin America.

"I know the bloodshed taking place around the Falkland Islands is of deep concern to every nation in this hemisphere," he said. "We understand and are sensitive to Latin American sympathies in this crisis, something which made our own decisions more painful and difficult."

"I hope you will, as neighbors and friends, do your utmost to understand the importance we attach to the principle that armed force should not be used to assert claims in an international dispute . . . . Let's make certain that emotions don't blur the truth of how close we really are during this tragic conflict."

Reagan's speech closely paralleled a conciliatory appeal by Secretary of State Alexander M. Haig Jr. Thursday at the opening here of the OAS foreign ministers' meeting. The meeting was called by Argentina in an effort to obtain the backing of Western Hemisphere governments for its struggle against Britain.

At Thursday's session, Argentine Foreign Minister Nicanor Costa Mendez bitterly criticized the United States for "turning its back" on Latin America, and sought to portray the Falklands fighting as a struggle between the Latin world and an "Anglo-Saxon" alliance of the industrial Northern Hemisphere.

The United States, while continuing to insist that its support for Britain is justified because Argentina was the first to resort to force, also has been very concerned that Latin American sympathy for Argentina will cause severe problems for future U.S. relations with the region.

Giving concrete form to that concern was yesterday's OAS session, which continued early this morning as a procession of Latin American ministers voiced strong support for Argentina and lengthy negotiations behind the scenes hammered out the resolution demanding that the United States stop aiding Britain and opening the way for OAS members to assist the Argentines.

The final language involved compromises that softened somewhat the tough language originally proposed by Argentina and its strongest backers: Peru, Venezuela and Panama. But it still deplored Britain's "unjustifiable attacks," it called on the United States to lift sanctions against Argentina and stop furnishing arms and other supplies to Britain, and it encouraged Latin American countries to assist Argentina in whatever manner they choose.

Argentina had sought the resolution under provisions of the 1947 Rio treaty of hemispheric mutual defense, to which 21 of the 30 active OAS members, including the United States, are signatories. On the vote, the United States, Chile, Colombia and Trinidad abstained.

The United States contends that the Rio treaty is not applicable because Argentina was the aggressor,and U.S. officials have argued that the resolution would encourage a prolongation of the fighting and impede efforts by U.N. Secretary General Javier Perez de Cuellar to negotiate a peaceful resolution.

U.S. sources, acknowledging privately that the resolution would not help U.S.-Latin American relations, said they had been encouraged by the positive response of many Latin delegations to the conciliatory tone of Reagan and Haig.

Except for a hard core of passionately pro-Argentine countries, the sources said, many told their U.S. counterparts privately that, while they would have to vote for the resolution, they understood the U.S. position.

Reagan, talking with reporters after his speech in California, was asked about reports that Britain wants a U.S. air base established in the Falklands. He replied:

"I don't know of any such proposal. I do know that a proposal is being considered for a cease-fire, that there has been a suggestion of a neutral peace-keeping force there while both sides withdraw and continue to negotiate--of which one of the participants would be the United States, just as we're one of the participants in the Sinai border force."