It's questionable whether an exultant Mario Cuomo can make the New York governor's race a referendum on Ronald Reagan and "the first battle of 1984." But there's no doubt at all that he has proven to dubious Democrats that it's okay to be a liberal, at least in New York.

Cuomo's Republican rival, Lewis Lehrman, is an ardent supply-sider who wishes to bring the blessings of Reaganomics to the Empire State. He hopes to make the contest a referendum on New York's retiring Democratic governor, Hugh Carey.

Cuomo's victory over New York's clamorous, histrionic, ever more conservative mayor, Edward Koch, was the most dramatic upset of the early season. To win, Cuomo, son of Italian immigrants, defied neoliberal chic, conventional wisdom, polls and the three New York City newspapers. He started out almost 50 percentage points behind Koch, trailed by 22 points on Labor Day and last Thursday beat him by six.

Cuomo hopes that national Democrats have taken in the fact that he came on as an unblushing progressive, spent less than half of what Koch did for television time and did not waver in opposition to the death penalty. He did it, he says, by running as "a real Democrat."

Cuomo warmed up for the Lehrman bout by working over Koch as a kind of northern "Boll Weevil," a Democrat who supported Reagan's cuts in social programs.

In one of his 30-second television spots, Cuomo showed Koch accusing Democrats of being too soft on the handicapped, unions and environmentalists. In another, he showed Koch receiving Reagan at Gracie Mansion in 1980 and 1981.

In the unexpectedly heavy primary vote, Cuomo won the women's vote. They may have remembered Koch's cavalier treatment of New York City Council President Carol Bellamy, whom he called "a horror show," and of former U.S. Rep. Elizabeth Holtzman, whose Republican Senate opponent, Alfonse M. D'Amato, was endorsed by Koch.

The minorities came out in record numbers -- the mayor's closing of inner-city facilities had given him a bad name with the blacks. And of course upstate voters, remembering Koch's airy comments in Playboy magazine about the "sterility" of country life, went for Cuomo by better than 2 to 1.

Cuomo, a street-smart intellectual, played heavily on New Yorkers' doubts about casting Koch as governor. He wore a "Keep Koch Mayor" button.

One New York Democrat said Koch's motivations for wanting to go to Albany were never quite clear: "There was a lot of talk about him being the first Jewish vice president, and people thought he was just looking for a steppingstone."

Cuomo's opposition to the death penalty should have cost him heavily in blue-collar constituencies but, in the end, economics counted more. Not even his own mother agrees with him about the death penalty -- one of Cuomo's daughters was attacked twice by the same man. But he said repeatedly in pre-primary debates that people were wrong to think "they can make their daughters safe with an electric chair."

Lehrman, like Koch, favors capital punishment.

Richard Wade, a Democratic activist and New York University professor, told Cuomo last January that he would win because the fight with Koch would be a replay of the Kennedy-Carter 1980 New York state primary struggle and other battles in which liberals prevailed.

Cuomo supported Carter, but Kennedy Democrats forgave him because Cuomo made it clear that he went with Carter only after Kennedy told Cuomo he would not go for the presidency. Kennedy came to New York several times and said nice things about Cuomo. Former vice president Walter F. Mondale, who endorsed Koch, has been telephoning Cuomo since last Thursday.

Cuomo is already calling Lehrman "a clone of Ronald Reagan" and deriding him as subscribing to the Reagan view that "if you take care of the rich, the rich will take care of the rest of us."

Lehrman, a newcomer to politics, is the retired president of his family's Rite-Aid drugstore chain. He spent $7 million, half of it from his own pocket, to win the primary. He is a tall, distinguished-looking candidate who, the Republican National Committee says approvingly, has run a model campaign, which they will aid in any way Lehrman suggests. Among his advisers, surprisingly, is Adam Walinsky, who as Robert Kennedy's speech writer penned some of the most memorable liberal documents of the 1960s.

Cuomo is calling for a "Siamese" campaign, joint appearances throughout the state and "in-depth discussions" instead of the 30-second spots that Lehrman can so much better afford to buy.

Lehrman, considered for a Reagan Cabinet post, plans to picture Cuomo as too far to the left for rank-and-file Democrats. And even though Carey supported Koch in the primary, Lehrman hopes to wrap the Carey record around the lieutenant governor's neck.

"The question is," Cuomo says, "whether you can associate yourself with Reagan, who is popular, without associating yourself with Reaganomics, which is not. Carey is not popular, but he has a good record. I hope the president comes in here. He'll help me make the point that there is a real difference between the two parties."