When state Sen. Phil H. Snowden is asked why he wants to put a balanced-budget amendment in the U.S. Constitution he is apt to hark back to some advice that Thomas Jefferson once gave for dealing with politicians:

"Hear no more of the faith of men but bind them down with the chains of the Constitution."

Adds Snowden: "That's what we should have done years ago."

An influential and respected member of the Missouri General Assembly, Snowden is one of many legislators across the country eager to use the Constitution to force Congress to start balancing budgets. His counterparts in 31 legislatures have already passed bills applying for a constitutional convention to do so; if only three more follow suit, the first convention since the Founding Fathers' would be theoretically in order.

But now a problem has cropped up for the budget balancers.

When Alaska became the 31st state to come into camp early this year there were at least half a dozen other states in which one legislative house had adopted a convention call. The expectation was that a sort of herd instinct would set in with legislatures tumbling over one another for the historic honor of putting the convention across.

Instead, the convention idea was suddenly defeated in the Washington state senate, bottled up in the Kentucky lower house by the speaker, and put into what one proponent called a "deep freeze" in Ohio. Missouri, where the senate passed it by a 2-to-1 ratio a few weeks ago, became a focal point in the drive.

And now even in this fiscally conservative state, the pro-amendment people have begun at least temporarily to lose ground. They admit the votes are slipping away and that only a major lobbying effort can turn things around.

Why the hesitation? A lot of Missourians started worrying about what they might lose if a constitutional convention requiring balanced budgets came to pass. Some fear it would mean a further reduction in federal aid on top of cuts already made for a state whose own budget is made up one third of federal funds.

Others sense danger in a convention that might wander into other fields. Liberals fear an assault on the Bill of Rights. Missouri's influential Baptists fear a convention might give federal aid to private, meaning parochial, schools. Many on both sides foresee a convention bogged down in explosive issues like abortion, handgun control and school busing.

Behind the campaign for balanced budgets is the propelling belief that Congress no longer controls the budget but merely votes the sum total of spending that special interests are able to gouge out of it. Snowden and others who favor an amendment picture Congress as a hapless mass needing help from outside, rather like, to use a metaphor that crops up frequently, a drunk who needs help to stay off the bottle.

"I don't know why we should expect this group in Congress to do it balance the budget ," Snowden says. "They've only done it once in 25 years. What gives them the incentive this year? They are structurally incapable of doing it and any congressman you talk to, if he's being honest with you, will say so."

The point is made succinctly by House Speaker Bob F. Griffin, who has endorsed the constitutional convention resolution as a means of giving Congress no legal escape from a balanced budget. "They need something to protect them from themselves," he insists.

Not even these ardent proponents think there will actually be a constitutional convention to propose such an amendment. Merely pushing it through the required 34 legislatures will force Congress itself to act first and propose the amendment, they contend. There is considerable sentiment in Congress for doing it already; the Senate Judiciary Committee has sent a proposed amendment to the Senate floor.

Without a sudden shift in sentiment, however, the amendment idea now does not seem likely to pass Missouri's lower house. A leading supporter, Rep. Gary Sharp, concedes that enthusiasm has diminished. Rep. Richard Hamilton, who once endorsed it but now has changed his mind, thinks it will die on a 5-to-5 or 6-to-4 vote in the Miscellaneous Resolutions Committee he heads. Speaker Griffin, who also favors it, states flatly the votes are lacking and says he will not call it to the floor at the tail end of a busy session for a meaningless debate.

Many factors are at work in the handsome statehouse that towers over the Missouri River here. Liberal organizations that fear the results of a constitutional convention--the AFL-CIO, the American Civil Liberties Union, the Anti-Defamation League and others--have lobbied hard this year to block it.

There also have been some sober second thoughts about the potential effects of a national budget forced suddenly into balance by about 1985. What, for example, would happen if federal spending and grants were suddenly cut sharply to fit the strictures of a balanced-budget amendment?

"People are worried about what would happen to federal grants to the states," acknowledges Hamilton, who has switched to a position opposing the measure. "They are starting to say, 'Well, government has been good, some good, to the people and we can't just shut down everything overnight.' It's going to take a more gradual thing, maybe 15 years down the road."

Snowden, the principal proponent here, shakes his head slowly and acknowledges that a fear of losing federal largess is in part responsible for the ground lost in Missouri. Yes, he agrees, there would probably be less federal money for states to divvy up and he recognizes that a lot of politicians don't want to accept the blame for that. "It's absolutely true but we just have to go ahead and take the blame," he says.

But the major force slowing down the momentum here is the sudden and unexpected opposition registered by the Missouri Baptist Convention, a powerful and conservative force in rural parts of this state. Its leaders are said to be concerned that a constitutional convention might stray into other issues and come up with an amendment permitting federal aid to parochial schools.

"The Baptist Convention threw us a curve ball," acknowledges James Davidson, chairman of the Washington-based National Taxpayers' Union, which has been the principal lobby in behalf of the amendment. "They opposed it on grounds that a convention would approve aid to Catholic schools. Now we have to lobby in the religious community to make up the lost ground."

This specter of a "runaway" convention, delving into many issues besides a balanced-budget amendment, is perhaps the main obstacle confronting convention proponents here. It has been suggested that the convention, if it came to pass, would take up such other controversial subjects as abortion, school busing, handguns and women's rights, as well as delve into the Bill of Rights.

Architects of the convention campaign contend that the convention would be restricted to the single issue proposed in the applications working their way through state legislatures, a balanced-budget amendment. They cite a study by the American Bar Association to support their case that a convention would be strictly limited.

They are opposed by a Duke University constitutional scholar, Walter E. Dellinger. He contends that the only type of convention Congress could call is one free to determine for itself what amendments to propose.

The proponents hope that in the legislature's closing days the fears of a runaway convention will be overcome by greater fears of runaway Congresses voting runaway deficits and bringing the country to ruin. To them, the problem no longer is one merely of spendthrift congressmen, profligate presidents and wasteful bureaucrats. It is a "structural" problem now, they argue, one that demands restraints imposed from outside.

Rep. Sharp warns of a spiraling deficit followed by deflated money followed by "the threat of a revolution and a complete overhaul of government," all because Congress cannot cope with the special interests any longer.

"We supposedly had a very conservative movement last year with a lot of pain and gnashing of teeth," Sharp recalls, speaking of President Reagan's budget-cutting strategy. "But I have no confidence in the ability of Congress to affect this, not when you look at the power of the special interests."