Leaders of liberal and labor organizations watched in frustration yesterday as many senators they are supporting for reelection this year voted for the balanced-budget amendment to the Constitution. In some cases, the angry lobbyists vowed to make them pay.

AFL-CIO President Lane Kirkland, calling the vote "the greatest act of political cowardice and chicanery in modern political history," said there would have to be discussions about how to deal with the Democrats who voted for the amendment.

One key labor political operator, Josiah Beeman, of the American Federations of State, County and Municipal Employes (AFSCME), said he would recommend a cutoff of financial aid to the candidates who supported the measure.

Eleven of the 19 Senate Democrats running for reelection--and 22 of 46 in all--voted for the amendment. At least half the candidates have received significant financial and political help from unions and liberal groups that were lobbying against the amendment.

But most of labor's anger was directed, not against the 1982 candidates, but against freshman Sen. Alan J. Dixon (D-Ill.), who was accused by Beeman, AFL-CIO lobbyist Ray Denison and others of having broken his word by voting for the measure.

Dixon's press secretary, Wade Nelson, said the senator "never committed publicly or privately" on the amendment, but the labor lobbyists insisted that Dixon had told Illinois AFL-CIO President Robert Gibson that he would vote against the proposal.

In an unusual move, the Democratic National Committee expressed its "regret" at the Senate action and said that despite the near-even split among Senate Democrats, the party was committed--by the action of its mid-term conference in Philadelphia in June--to opposing the constitutional amendment.

Gene Eidenberg, the director of the DNC, who expressed the "regret," said, "It's my hope the Democratic leadership in the House will be able to withstand the pressure" and defeat the amendment there.

The split among Senate Democrats began at the top, with Minority Leader Robert C. Byrd (D-W.Va.), voting for the amendment at the end of the roll call, while his deputy, Sen. Alan Cranston (D-Calif.), led the fight against it.

Cranston told reporters that the vote "does not reflect adversely" on Byrd's leadership, but there were rumbles among other senators and some labor lobbyists that it would be remembered.

Byrd has received at least $147,250 from labor and education groups for his reelection campaign. He said in a statement after the vote that objections to the amendment "trouble me greatly," but he decided that "a question of this magnitude" should be put "directly to the people" by sending the amendment to the state legislatures for ratification.

Other 1982 candidates with major labor and education funds who voted for the amendment yesterday were Sens. Howard W. Cannon (D-Nev.), Quentin N. Burdick (D-N.D.), John Melcher (D-Mont.) and Jim Sasser (D-Tenn.). Most of them had endorsed the amendment months ago and their positions were not a surprise to the labor-liberal lobbyists.

Nonetheless, Beeman said, "I think our AFSCME members will have a great deal of difficulty supporting people who, in effect, voted to abolish their jobs . . . . I certainly wouldn't recommend any additional contributions to them."

The Democrats were put in even worse light by the fact that three Republicans running for reelection with significant labor support--Sens. John Heinz (Pa.), John H. Chafee (R.I.) and Lowell P. Weicker Jr. (Conn.)--resisted strong White House lobbying pressure and voted against the amendment.

The only labor-backed Republican to support the amendment was Sen. Robert T. Stafford (Vt.).

Kirkland told reporters at the AFL-CIO executive committee meeting in New York that the labor federation would mount a major campaign to defeat the amendment in the House, and, if that failed, to deny ratification in the states.

The latest setback to labor and its Democratic allies came on the day that union groups and congressional Democrats were conducting demonstrations and making speeches denouncing the economic record of the Reagan administration on the first anniversary of the passage of the 1981 budget and tax bills.