The national campaign for a balanced-budget constitutional amendment, one of the brightest comets in America's political skies just a few months ago, has flamed out - at least for this year.

With most state legislatures in adjournment, or close to it, it now seems clear that balanced-budget proponents will not succeed this year in their effort to win the 34 state resolutions necessary to call a constitutional convention to propose a budget amendment.

On Capitol Hill, meanwhile, subcommittees are moving at a glacier's pace in their consideration of numerous proposals for an amendment. It is doubtful that any such proposal will emerge from committee in either house this year.

The movement's long-term prospects are equally unclear. The National Taxpayers Union, the Washington-based lobby that coordinated the drive, still predicts that a constitutional amendment will eventually be proposed, either by Congress or a convention. But some of the group's allies now say the drive is likely to fall well short of its goal.

All this happened to a movement that has gained lopsided public support (polls three months ago showed more than 75 percent of the American public favoring a budget amendment), a broad grass-roots organization, and the active backing of a major politicial figure, California Gov. Edmund G. (Jerry) Brown Jr.

What happened? To a degree, the amendment drive has been a victim of its own success. As it gained public attention, it spawned a counterattack from the Washington political establishment.

Further, the balanced-budget campaign has become involved in all sorts of extraneous political battles including the competition between Brown and President Carter for the Democratic presidential nomination next year.

Both supporters and opponents of the balanced-budget push agree that Brown's support has been a mixed blessing at best. The Californian helped give the issue national prominence, but he also gave it a flavor of political opportunism. "We have to explain to everybody that we're not running Jerry Brown's presidential campaign," James Davidson, head of the Taxpayers' Union, complained in March.

The taxpayers union, working with a scattered corps of state legislators, set out four years ago in an uphill effort to convince Congress to propose an amendment mandating a balanced federal budget, or to force a convention to propose such an amendment.

Constitutional amendments can be proposed by Congress or by a convention called at the request of two-thirds (34) of the states. In either case, a proposed amendment must be approved by three-fourths (38) of the states to take effect.

By the beginning of 1979, 22 states had passed resolutions calling for a convention. In the first four months of this year, eight more states climbed aboard. (Of those 30 resolutions, however, at least six have legal or contextual difficulties that might render them invalid.) It was widely predicted that the necessary 34 resolutions would be in hand by this June.

That rush of success drew considerable attention to the campaign. The attention was increased exponentially by Brown's announcement, in January, that he would campaign around the country on behalf of a constitutional convention.

All the attention, however, prompted a new sense of caution among state legislators. Convention resolutions - which used to breeze through legislatures with almost no debate - now prompt tough, extended battles. "It was one thing when you could just pass the thing and send it off to Washington with nobody looking," says Ohio state Sen. William F. Bowen. "But now the newspapers are watching, you've got to have hearings. Everybody's more careful when this comes up in a legislature now."

As the possibility grew that 34 states might call for a convention, political leaders in Washington swung into action to head off the drive. To answer the charge that Congress would not act on the amendment proposal until it was forced to, committees in both houses began extended hearings on various constitutional and legislative limitations on the federal budget.

Congress also passed a provision requiring the Budget committees in both houses to come up with a plan for balancing the federal budget in fiscal 1981. Although this action was denounced by the strongest balanced-budget advocates as a hollow gesture - the plans are purely advisory - it created the impression that Congress was getting serious about a balanced budget.

Presidential politics, too, have played a part. When Brown entered the fight for a constitutional amendment, Carter set up a White House task force to work against it. With some nudging from Democratic party leaders in Washington, Massachusetts Lt. Gov. Thomas P. O'Neill III set up an organization called Citizens for the Constitution to combat Brown and the balanced-budget drive.

As a result, the debate in several states has been determined at least as much by political allegiance as by principle. In the New Hampshire legislature, the struggle over the convention resolution turned into an open battle between Carter supporters and Brown's backers. (Brown won.)

In Ohio the resolution has become a pivot point in a contest between Carter loyalists and dissident Democrats who want to draft Sen. Edward M. Kennedy (D-Mass.) for the presidency.

Despite all these difficulties, the taxpayers union says it is still confident that it will eventually force a convention to propose a balanced-budget amendment. "We've got 30 states now," says George Snyder, the group's hard-working campaign director. "That means we need four more, and we've got 20 to shoot at."

O'Neill's anti-amendment group recognizes that mathematical fact, but doesn't know exactly what to do about it. O'Neill himself is confident that some legislatures that approved a convention resolution a year or more ago would change their minds if the issue came up again. But he cannot push for recision of state resolutions, because that would cause trouble for another liberal constituency, the women's rights movement, which is contesting the legality of state recisions of resolutions approving the Equal Rights Amendment.

The future of the amendment drive - a movement that gets its strength from national frustration over inflation - may well depend on the economy. If the nation heads into an economic slowdown, or recession, later this year, the balanced-budget campaign, for all its past successes, could easily run aground over the long haul.