Many Democrats face a dilemma in choosing between Walter F. Mondale, the old friend with whom they have waged frequent battles for liberal programs in the past, and John Glenn, the newcomer whose policies are unclear but whose voter appeal is obvious.

Few Democrats feel that perplexity more acutely than Gov. Mario Cuomo of New York, the bright new star of the party and likely convention keynoter. His endorsement will help determine the 285 votes of the second-largest delegation.

Cuomo bought some time for reflection by arranging a series of autumn "issues forums" around the state for the seven presidential hopefuls. But the last of those forums will be held on Oct. 6. And some time in November, Cuomo knows, he will have to make his choice.

The 62 county chairmen and virtually all the major elected officials have stayed neutral, out of deference to the governor. But Cuomo understands that they cannot be asked to wait beyond Thanksgiving Day.

He might stay neutral, or seek a favorite-son status in the April primary, but that goes against the grain. It implies that he finds none of the candidates worthy of support or that it is not in his personal interest to back the one he considers best. That kind of cynicism or selfishness offends the governor's self-image of idealism.

But he is practical enough to know that he cannot take New York's leadership with him for any of the long shots, no matter how estimable they might be. The party would splinter and he would look weak. So it must be Mondale or Glenn.

Mondale could make it easy for him--and for others around the country who share Cuomo's commitment to the Democratic Party as the instrument of social justice and economic opportunity. If only he would speak and act like a bold leader, not a front-runner nervously sitting on a lead, the endorsement decision would not be in doubt.

But when a person whose insights mean much to Cuomo says, after watching Mondale in a television interview, that he seems a man already weary of the struggle, meek of manner and of voice, and such men cannot win the White House, it gives the governor real concern.

It reinforces the doubts Cuomo nurtures--despite his efforts to put them aside--about Mondale's equivocation in Cuomo's 1982 gubernatorial primary with New York City Mayor Ed Koch. Mondale had every reason of principle and political loyalty to back Cuomo, the liberal in the race and the strongest ally of the Carter-Mondale administration in New York state politics. But Koch was the strong early favorite, and if Mondale did not implicitly ally himself with Koch, as some suggest, he certainly did nothing to aid Cuomo.

Yet Cuomo won that fight--and won it by sticking to his principled opposition to the death penalty, when even his own mother thought it would cost him victory. Is there not some issue that means as much to Mondale, some principle he would defend even at the risk of affronting an important constituency? And if there is not, what does it say about the man?

Glenn is something else. His public appeal is obvious, even before he speaks. Cuomo has seen supposedly sophisticated New Yorkers--immune to ordinary political glamour--press close for Glenn's autograph or his touch. He is a classic American-- a Marine hero, an astronaut, a man who plainly commands his own mind and emotions and will.

And yet that strong streak of independence and self-reliance must be at least a bit disturbing to a governor who saw close-up how much the inability to work closely with other politicians cost another loner, Jimmy Carter, in the presidency.

There are disturbing signs about Glenn: conversations where the senator appears not to listen to others speaking, not to grasp what they are saying, not to incorporate their ideas into his own or show how their separate approaches can be merged. Such men, even when they are right on an issue, have difficulty building majority coalitions. And senators Cuomo knows have told him Glenn has exactly that kind of difficulty in the Senate.

And what does Glenn really believe? In 1979, he opposed the SALT II treaty. In 1981, he supported Reagan's tax package. Just recently, he voted to resume nerve gas production. To some, he seems ambivalent about Israel. What does this tell a governor of New York, who wants no watered-down Reaganism in the Democratic Party, about the implications of a Glenn presidency?

Why doesn't Mondale concede publicly that Glenn has it all over him in a personality contest and demonstrate his own political courage by challenging Glenn directly on these issues? Why not confront the waverers in the Democratic electorate by asking them--as Cuomo asked New York Democrats in the 1982 primary--to vote not on the basis of polls or endorsements or "smart money" shrewdness, but on the basis of their deepest beliefs about what the Democratic Party should stand for in America? Why can't the choice be made clear? Mario Cuomo--and many like him--would like to know.