Organization is key in low-turnout N.Y.

New Yorkers are pleased that next Tuesday's primary is seen as such an important testing ground for the hopes of the Democratic presidential aspirants. ''There is a great sense of excitement,'' says a Walter F. Mondale supporter. Because the race is so close, ''people are now in a crisis stage. There is a lot more enthusiasm.''

''This is the most exciting race in 25 years,'' says a Democrat who plans to vote April 3 for the Rev. Jesse Jackson. He adds that New York delivers far more delegates than New Hampshire or Iowa. ''We should be listened to.''

The three contenders seem to agree. Over the past several days there has been a whirlwind of activity by all three camps. Mr. Mondale spoke at a labor rally, preached at a Harlem church, and visited New York City's South Street Seaport. Mr. Jackson flew to Buffalo and then on to Rochester upstate. Gary Hart spoke amid noisy subway trains in Queens and made several trips upstate.

A poll in USA Today Monday showed Mondale a seven-point favorite over Hart, but 11 percent of those polled were undecided.

Economic concerns top the list of issues in New York, say staff members from all campaigns. Environmental issues, such as acid rain, are important as well. But New Yorkers are also ''extremely sophisticated'' on foreign-policy issues, Ellen Chesler of Hart's New York staff points out. Questions on the Middle East, Central America, and arms talks are often fielded.

Between the two front-runners, this has sometimes translated into a fight over who is more loyal to Israel. Each hopes to capture New York's large Jewish vote.

Melba Tolliver of the Jackson campaign says New York voters are interested in the same issues most urban centers face: jobs, health care, housing, and concern over the arms race and military spending. Upstate, she says, Jackson will also focus on factory closings and the demise of smokestack industries.

It remains to be seen whether the competitive Democratic race will draw New Yorkers to the polls. The state has a record of low primary turnouts, says Roman Hedges, an associate professor at the Rockefeller College at the State University of New York in Albany. Roughly one-third of the voters have come to the polls for primary elections in the past, Dr. Hedges says. There was an increase in voter registration before the primary, but nothing ''abnormal'' compared with most presidential election years, says a spokeswoman at the state board of elections.

Because nearly two-thirds of the state's registered Democrats live in New York City, it dominates the election, even when voter turnout is low.

''The conventional wisdom is that those who can come to N.Y.C. with a split vote and have appeal (elsewhere in the state) are going to win,'' says Hedges.

Although the strong endorsement of organized labor may have hurt former Vice-President Mondale in some states, it could be a mighty advantage in New York. In a state with a record for low turnout at primaries, there is clearly a premium on organization, observers say. It is not so much what percentage of labor votes for which candidate, but what kind of effort labor can offer to mobilize voters. At a labor rally this past weekend, AFL-CIO state president Edward J. Cleary promised support for Mondale through direct mail, phone banks, and one-on-one contact with union members.

''If you've got an organization with a phone bank and that can help to get people to the polls, it is a tremendous advantage,'' Dr. Hedges says.

Though Mondale has the support of top party officials in New York, including Gov. Mario M. Cuomo and Mayor Edward I. Koch, spokespersons from both the Hart and Jackson campaigns are confident that they are bringing some change to New York politics.

''There is a whole crop of people who are earning their spurs here,'' says a field organizer for Hart at the frenetic New York City headquarters. He reports more than 4,500 volunteers throughout the state. ''For some it is their first brush, or a return to politics, and it is a very positive experience. They will move into politics somewhere down the line.''

''This is something that will continue beyond November,'' says Ms. Tolliver at Jackson headquarters. In fact, the question of whether Jesse Jackson can play Pied Piper to minority voters, particularly in New York City, could most affect future politics here, observers say. There is an ''extraordinarily volatile'' black vote in New York City, says one, that has been polarized in recent years. If Jackson is able to mobilize it, the vote could potentially upset city politics.

Some black voters are very aware of this potential, both at the local and national level.

''In the primary I will vote for Jesse, because he'll send a message,'' says Walter Williams, a transit worker from Brooklyn. He supports Mondale over Hart for the November election.