Unlike President Reagan, Gov. Mario M. Cuomo of New York is taking bows for his budget.

He is being praised for spreading the pain among the citizenry. He called for a reduction of 14,000 state employes, for taxes on alcohol, cigarettes and excess profits of real estate and oil companies.

Again unlike Reagan, Cuomo won wide support for his balanced budget by an exhaustive round of all-inclusive consultations and negotiations.

Within its austere confines, he made provision to expand facilities for the homeless and to ease overcrowding in the state's prisons.

Cuomo says he hopes this display of decency and equity will not be lost on the several Democrats who wish to be president. He makes no bones about wanting to have a say about the nominee and the Democratic platform.

As the most important Democratic governor in the country and the leader of the convention's second largest delegation, he is bound to be heeded.

The Democratic National Committee apparently is glad to acknowledge his preeminence, and has invited him to address its May 14 strategy council. The presidential hopefuls call on him frequently in New York. Until he dropped out, Sen. Dale Bumpers (D-Ark.), a breakaway risk-taker after Cuomo's heart, was in constant touch.

Cuomo already has established himself as his party's premier stylist. A prideful fellow Italian-American calls him "an ethnic Adlai Stevenson."

His inaugural address was an elegant and compassionate manifesto in which he stated his credo that government "can pay the bills and still help people in wheelchairs." His speech to a celebration dinner attended by all Democratic contenders except former vice president Walter F. Mondale, who announced that he was busy working on his announcement statement, so impressed Sen. John Glenn's (D-Ohio) wife, Annie, that she asked the governor for the name of his speechwriter. She hoped to engage that person for her husband.

"You'll have to come to Albany to get me," said Cuomo, who writes his own.

Just how Cuomo can influence the presidential campaign is not quite clear. One of the surest signals that he hopes to make a difference was his hiring of Timothy Russert, the ace who served Sen. Daniel Patrick Moynihan (D-N.Y.) as his liaison to Washington and the press.

One way they devised to make the presidential race seem wide open was to persuade New York Democrats to withhold endorsement until after all claimants have appeared before a series of state-wide forums in September.

Cuomo doesn't like to hear that the choice is "inevitable."

The other night on his way to an Eastern Queens Democratic Club festival in honor of Queens County District Attorney John Santucci, the governor said, "One thing I'm sure of. Before it's over four or five things will happen that nobody can foresee, something like Romney saying he was brainwashed or Muskie crying."

He muses, for instance, on what will happen when Glenn goes south, or when a movie about Glenn's astronaut days, "The Right Stuff," is released this fall.

"I hear there's one terrific scene that shows Glenn putting his foot down when Lyndon Johnson then vice president tries to arrange to be shown in Glenn's house with Annie Glenn--she stuttered, you know, and he wants to protect her, even though it could cost him his place in the mission."

Cuomo is close in liberal ideology to front-runner Mondale, whom he has known the longest and the best. But there is a reserve in the relationship, having to do with Mondale's performance in the New York Democratic primary, in which Cuomo was the underdog. Mondale never officially endorsed Cuomo's heavily favored rival, New York Mayor Edward I. Koch, but, according to Cuomo, "He might as well have--his money people went with Koch and he didn't go with me."

In 1980 Lt. Gov. Cuomo bore the heat of the day as captain of the Carter-Mondale forces, while Koch played both sides: "nominally" endorsing the Democratic ticket but inviting Reagan to Gracie Mansion on Labor Day. Cuomo obviously feels that he deserved better than Mondale's neutrality.

"I have no personal difficulty in embracing the idea of a Mondale candidacy," he says. "It's just too early, and I resent the assumption that the favorite should always win."

In New York, among his own, Cuomo is forgiven his intellect, his wit and his stand against the death penalty because he has not gotten above himself. He grew up over a Queens grocery store owned by his Sicilian immigrant parents. At his inaugural, he ad-libbed a personal note: "Pop, wherever you are--and I think I know--please don't let me forget."

At the Santucci dinner, he said, "We all take potshots at politicians. We enjoy running ourselves down. But politics is the grandest and noblest profession: it is the business of trying to improve people's lives."

That's what he wants to hear from the Democratic nominee.