But the governor has remained loath to go after Mr. Trump by name, even as the president’s popularity has plunged to historic lows, and as other Democrats, who signaled an early willingness to work with the president, like Senator Chuck Schumer, have adopted a harder line. Hours after Mr. Cuomo held a rally against the Republican health care bill last week in Manhattan and did not mention the president’s name in his 15-minute speech, Mr. Schumer branded that day’s setback a “failure of Trumpcare.”

In June, when Mr. Cuomo wrote an op-ed lamenting the Trump administration’s immigration policies, he blamed the federal government, not Mr. Trump, for having “forgotten who we are as a nation.” In July, when Mr. Cuomo wrote about the health care bill, he cited “the White House’s insistence” that the legislation had “heart,” though it was Mr. Trump himself who had said so.

And when reporters recently asked Mr. Cuomo about Donald Trump Jr.’s email with a Russia-linked lawyer, he took a pass. “I’ve been working,” he said. Mr. Cuomo also skipped commenting on Mr. Trump’s Twitter attacks on the MSNBC host Mika Brzezinski. “Let me read them first,” he replied.

The nonanswers sounded more like the comments of congressional Republicans who have grown skilled in the art of the Trump dodge than those of a nationally ambitious Democrat.

In contrast, at the recent rally where Mr. Cuomo left the president unnamed, the Democratic state attorney general, Eric T. Schneiderman, introduced himself “as a guy who sues Donald Trump.” It earned some of his loudest applause.

Mayor Bill de Blasio of New York City, a Democrat, traveled to Germany this month in hopes of casting himself as a foil to the president. And Senator Kirsten E. Gillibrand, Democrat of New York, who also could be a 2020 presidential contender, has been buying up online search ads calling herself a leader of the “resistance” who is “standing up to Trump.”

“Cuomo could, if he wanted, attack Trump 20 times a day and it wouldn’t hurt him a bit in the state of New York,” said Robert M. Shrum, a longtime national Democratic strategist. Mr. Shrum said the governor had demonstrated “a habit of not getting involved in national politics,” though he added, “I’m not saying this would be my strategy.”