NEW YORK, DEC. 10 -- The six Democratic candidates for president found out tonight what millions in lesser circumstances have known for years: The Big Apple can be one tough town.

At a forum sponsored by the New York State Democratic Party, they spent the evening fielding skeptical questions about their electability and leadership before an audience of several thousand Democrats who -- judging from the absence of buttons and tepidness of applause -- aren't ready to give their hearts to any of them.

They weren't even publicly welcomed by their host, New York Gov. Mario M. Cuomo, who remained in the audience, aides said, out of fear that if he gave welcoming remarks he would be accused of trying to upstage a field he has chosen not to join.

The tone of the 90-minute session was set by the first question, which was from local radio commentator Sherrye Henry. After rattling off poll numbers showing that "undecided" continues to lead the Democratic race, she asked, "Why are so many Democrats not pleased with their choices?"

None of the candidates responded directly, but the one who drew the best audience reaction to that question -- as he did throughout the evening -- was Jesse L. Jackson.

He issued a "moral" challenge to voters to consider his candidacy without considering his race. And he said that if he could be elected, no longer would "sex, race, religion and an insatiable military appetite be a prerequisite" for the presidency.

Jackson also drew sustained applause here -- in a city that he called "Hymietown" during the 1984 campaign -- with a quip about the Intermediate-Range Nuclear Forces Treaty signed this week in Washington. He called it "a small step for mankind and a giant step for Reagan."

"I wish Jesse Jackson could be the coach for all of them," said William Haddad, a longtime New York Democratic activist who is supporting Sen. Albert Gore Jr. (D-Tenn.). "He's got the ability to crystallize things in a phrase."

The other candidate who had a lively evening was former Arizona governor Bruce Babbitt, whose best moment came when he simultaneously poked fun at Massachusetts Gov. Michael S. Dukakis' program to collect more taxes and Illinois Sen. Paul Simon's plan to enact a New Deal-style jobs program.

After questioning the cost, revenue estimates and effectiveness of both programs, he deadpanned: "We should take Paul's $10 billion {for the jobs program} and use it to hire Mike's 30,000 new tax collectors."

Babbitt also reenacted his "stand up" for higher taxes -- a stage gimmick he unveiled at the nationally televised NBC debate last week. This time, instead of challenging his opponents to join him, he asked the audience. There was some applause, but only a few among thousands stood up.

"Well, I didn't get you all," the somewhat dispirited Babbitt said. "I know New York is a tough sell . . . . But I'll be back."

At one point in the evening, after the candidates gave their standard answers to questions about taxes and deficits, comoderator Bill Moyers wondered aloud whether any of them had shown the "moral and philosophic" breadth that voters are looking for.

Babbitt shot back that when he had tried to speak honestly about the need for tax increases, Moyers had accused him of having "a death wish . . . . It's part of your job to egg us on," he chided the veteran journalist, "rather than push me off a cliff."

But Rep. Richard A. Gephardt (D-Mo.) dismissed the idea that the electorate is looking for rhetorical flourishes from the next president. "Everybody is from Missouri now," he said of an electorate that has been battered by Watergate, Vietnam, the Iran-contra scandal and record budget deficits. "Everybody says 'Show me. Don't tell me; show me.' "

The party's leading rhetorician, Cuomo, had kind words for everyone afterward. "The growth continued tonight. And it will continue until a star is born." He said he has not decided when or whom to endorse. "I clapped the same number of claps for them all."

Cuomo has recently been stung by criticism that he has been overshadowing the candidates. He has responded by lowering his profile and cutting back on travel. In recent weeks, he has canceled trips to, among other states, Iowa and New Hampshire. He reiterated tonight that he does not expect his party to draft him.

This week's summit and arms-control agreement received only fleeting attention in the forum. All of the Democrats had previously endorsed the INF Treaty and they vied only over who would follow up the opening most quickly and effectively.

Jackson called for television coverage of the missile destruction to dramatize the appeal of disarmament. Simon said he would challenge Soviet leader Mikhail Gorbachev to stop all nuclear-weapons testing immediately. And Babbitt said he would invite the Soviets to join the United States in a a Third World development fund. Gore, who has specialized in arms-control legislation, said it is "vitally important to have a president who can match" Gorbachev's mastery of those issues.

Gore and Dukakis appeared to strike a responsive chord with the New Yorkers when they asserted that the problem of homelessness will not be solved until President Reagan's cuts in federal housing programs are reversed. And Gore, who has shown few signs of strength in New York, was cheered when he said, "Yes. Absolutely," the federal government should share New York's estimated $1 billion annual cost for treatment of people with AIDS.

Simon, who seemed content once again with a modest debate role, drew his strongest cheers when he said that New York and other big cities depend on federal aid and that he is one Democrat not so worried "about being labeled a big spender" that he is "not willing to make the necessary investments" in jobs, education and housing.