New York Gov. Mario M. Cuomo (D) shifted to more subtle weapons yesterday in his duel with Vice President Bush, suggesting that Republicans would be smart to give the 1988 presidential nomination to a key Bush backer, Treasury Secretary James A. Baker III.

Cuomo, who has been engaged in harsh verbal battles with Bush, downplayed the struggle, but at the same time downgraded Bush. "I don't think Bush . . . Bush is obviously the front-runner, he's the vice president, he's the logical candidate, I'm sure he's a good man," Cuomo said. Last month he called Bush a "menace."

In a rambling luncheon session with reporters, Cuomo said: "I'll give you another name on the Republican side I hope the Republicans are not smart enough to do . . . Jim Baker. . . . He would be a formidable candidate."

Baker, Bush's 1980 campaign manager and first chief of staff at the Reagan White House, said in reply: "I have a candidate for 1988, and everybody knows who it is."

Marlin M. Fitzwater, Bush's press secretary, initially said: "I don't think we'll get involved in that kind of speculation. Thank God the Democrats are not picking the Republican candidate anyway. Mario Cuomo is a . . . " -- Fitzwater paused -- "I better not comment. Let him have his fun." A few minutes later, Fitzwater called back with a revised response: "Jim Baker would be a good candidate."

In addition to promoting a James Baker candidacy, Cuomo said the key issue in the 1988 race will be "competence" and that Senate Majority Leader Robert J. Dole (R-Kan.) could establish his competence "overnight."

If Dole could pilot the Senate to a significant cut in the federal budget deficit, Cuomo said, he "could say 'the rest of them are making speeches, I'm making policy. The rest are speaking in poetry, I'm speaking in prose.' "

Cuomo professed little interest in the presidency. "I don't get any sense at this stage in my life that the country needs me. . . . I don't get any sense that you need a Cuomo." He was quick to note, however, that "I didn't say I don't want to be president."

While keeping his prospects alive, he in effect declared that another often-mentioned potential candidate, Chrysler Corp. Chairman Lee A. Iacocca, will not run. He said Iacocca told him, "I don't think I could go out there and take this kind of garbage that they throw at you. I don't need it; I don't want it."

Cuomo touched on a number of other subjects.After House passage of tax-revision legislation last year, he said, Ways and Means Committee Chairman Dan Rostenkowski (D-Ill.) "agreed that if the thing came back to a House-Senate conference committee, he would never allow even a compromise on the issue of deductibility" of state and local taxes. The governor was a leader in the fight to preserve the deduction. Cuomo agreed with Democratic National Committee Chairman Paul G. Kirk Jr. that the party's 1988 ticket is very likely to include a southerner or westerner. But he said the reason is that so many people from those regions are running, not that such a nomination is the only way to win in southern or western states. He predicted that the issue of the poor, or the "underclass," will reemerge in national political debate. He said a Republican group gave him a standing ovation recently when he said this about poverty: "One out of seven people were slaves when Lincoln was president, but now one in seven people have a new kind of enslavement, and Lincoln's work is not done."