With the backing of a Democratic labor leader, a nationwide direct-mail fund-raising campaign is being organized to draft Sen. Edward M. Kennedy (D-Mass.) for president.

The drive is being handled by a Washington political mailing firm, which also raises money for the Democratic Party. The first fund-raising letter is signed by William W. Winpisinger, president of the Machinists Union and a leader of the Party's liberal wing.

Winpisinger, an outspoken critic of the Carter administration, said the Kennedy draft is being organized because the government under President Carter "is hopelessly paralyzed. It flounders hopelessly, without direction in its approaches to the pressing problems of today."

Winpisinger conceded that Carter "is an honest, dedicated public servant with good intentions." But he also said "that is simply not enough" and added that Kennedy "is the one who can provide . . . true leadership."

Roger Craver, president of mailing firm, Craver, Mathews and Smith, said 20,000 to 50,000 letters will be sent out beginning next week. He said returns from the mailing will be held until Kennedy either announces for president or takes himself out of the race.

Kennedy has repeatedly said he is not a candidate and that he expects to support Carter for reelection. However, the senator also has declined to make any solid statement that he will not become a candidate at some time in the future. And Kennedy has done nothing directly to stop the various draft movements that have sprung up across the country in recent weeks.

"Considering the alternatives facing us, you and I simply must find a way to convince Sen. Edward Kennedy to run for the presidency next year." Winpisinger says in the letter, which asks for contributions of $15 to $100 or more.

Winpisinger acknowledges Kennedy's reluctance to challenge Carter for the Democratic nomination.

"Nevertheless, we are firmly convinced that he would run - under one very special condition," Winpisinger says in the letter. "And that is, if there is a genuine, common, non-political, grass-roots, people's draft calling on him to announce as a candidate for the presidency."

William Holayter, the Machinists political director, said there are about 17 charter members of the movement, called the "National Call for Kennedy."

The group has been incorporated in the District of Columbia with the incorporators listed as Victor Kamber, an officer of the Building and Trades Council of the AFL-CIO; Jules Bernstein, a Washington labor lawyer; and Sandy DeMint, executive director of the National Resource Center for Consumers of Legal Services.

Craver said the group is not registered with the Federal Election Commission as a political committee because it is not directly contributing to any campaign. But he said it will follow FEC rules and not accept any gift over the $1,000 legal limit for a campaign contribution.

Any money left over from the drive will be contributed to the John F. Kennedy Presidential Library, he said.