NEW YORK, APRIL 14 -- Randy Miller, a gay activist, decided to join Jesse L. Jackson's presidential campaign after hearing Jackson speak to the 200,000 gay-rights advocates who marched in Washington last October.

"It was a very powerful moment for me," said Miller, 29, who attended Howard University and earned a divinity degree at a small Virginia college. "Jesse Jackson has been speaking out for gay and lesbian rights. He doesn't just say it in New York City; he says it in Iowa."

When Jackson addresses gay men and lesbians at a Greenwich Village rally Friday, it will be another brief moment in the sun for a group that has traditionally had little clout in national politics. All three Democratic presidential candidates are courting New York City's gay community, a significant bloc in the state's April 19 primary. But Jackson has made the most emotional connection.

Massachusetts Gov. Michael S. Dukakis has picked up some gay support here, most recently in an editorial by the New York Native, the city's largest gay newspaper. Sen. Albert Gore Jr. (D-Tenn.) has been endorsed by one small, more conservative gay group.

But several gay leaders here are sharply critical of Dukakis' record on gay issues, with some calling him "homophobic."

"Dukakis is someone who has gone out of his way to hurt us," said David Taylor, president of Manhattan's Gay and Lesbian Independent Democrats, a 400-member club that has endorsed Jackson.

The contest for gay voters is almost a microcosm of the larger campaign: While Jackson moves gays with his eloquent speeches on gay rights, Dukakis takes a more measured approach and finds himself pinned down on specifics from his tenure as governor.

Several gay-rights activists criticize a Dukakis policy that allows gay couples in Massachusetts to have foster children only if no placement with a traditional family can be found. They assail his support of state rules that allow insurance companies to require AIDS tests for individuals seeking life-insurance policies.

Gay-rights activists also fault the governor for supporting mandatory AIDS testing in the armed forces and for failing to rescue a gay-rights bill. Timothy McNeill, associate issues director for the Dukakis campaign, said Dukakis is "very strong" on gay issues and that critics have focused on minor details. He said the state strictly regulates insurance-company AIDS testing and that Dukakis supports voluntary testing in almost all instances, with the exception of the military.

At a debate here Tuesday, Dukakis said his foster-parents policy "makes sense" and stressed that "we don't prohibit the placement of children in gay and lesbian households." But Virginia Apuzzo, a New York consumer official who is Gov. Mario M. Cuomo's liaison to gays, said that Dukakis' stance indicates "a very fundamental homophobia" that is "feeding into the most fundamental myth about gays and lesbians," that they cannot be trusted around children.

Jackson drew cheers from a vocal gay contingent at the debate when he spoke of the AIDS "hysteria." Recalling the march on Washington, he said: "I saw people in their wheelchairs who are dying of AIDS. . . . Not one of the {Reagan administration} officials would come downstairs and shake their hand."

Nearly every activist interviewed recalled how Jackson had mentioned gay men and lesbians in his speech to the 1984 Democratic National Convention. "People have never forgotten that," Taylor said. "We're usually unmentionable."

Still, gay-rights activists say they would back either of Jackson's Democratic rivals against Vice President Bush, who is part of an administration that is seen as having been slow to respond to the AIDS crisis. At Tuesday's debate, the candidates said there is a need for increased AIDS research and voluntary AIDS testing, although Dukakis and Gore would make isolated exceptions.

"In the gay and lesbian community, mandatory testing is seen as kind of a litmus test for the right to privacy," Miller said.

Apuzzo said that many less-vocal gays are "closet Dukakis supporters" who like the governor's experience and see him as the party's best hope for defeating Bush.

"We're looking for a candidate who can win," said Charles Ortleb, editor of the Native. "We can't just vote as purists in this election."

The worst alternative, Ortleb said, is continuation of a Republican administration that he described as "just deadly to the gay community -- a combination of incompetence and murderous intent with regard to the {AIDS} epidemic."

Henry Weiss, a politically active Manhattan lawyer who backs Dukakis, said gay Democrats are divided. "I don't think the gay community will vote as a bloc," he said. "We haven't been able to deliver for our own candidates, let alone someone who's running for president."