If Mr. Cuomo ran for the White House someday, it would be a remarkable turnaround for a man whose career had been written off on more than one occasion, nearly succumbing to political miscalculations. It would also offer the intriguing story line of a son’s fulfilling a mission endlessly contemplated, but never executed, by his father, former Gov. Mario M. Cuomo, who was also a Democrat.

Political consultants note several factors in the governor’s favor, including his ability to raise money: Mr. Cuomo already has a re-election war chest of nearly $20 million. He has also shown a willingness to blast Washingtonian dysfunction and Trumpian values, announcing shortly after the election in a fund-raising email that “as New Yorkers, we have fundamentally different philosophies than what Donald Trump laid out in his campaign,” something he echoed in his remarks on Monday.

“Let the great State of New York serve as a safe harbor for our progressive principles and social justice that made America,” the governor said.

There are other possible pluses, strategists say, including his age, 59; his religious background as a Catholic, potentially popular in Rust Belt states; and his current occupation as governor, an executive-branch position that has allowed politicians to rack up résumé-building accomplishments. The role, however, can come with unflattering headlines.

Preet Bharara, the United States attorney for the Southern District of New York, has twice investigated the governor’s office, and brought down two of Albany’s most powerful politicians: the former State Assembly speaker Sheldon Silver and the former Senate majority leader Dean G. Skelos, both convicted of federal corruption charges in late 2015.

While Mr. Cuomo has never been accused of any wrongdoing, Mr. Bharara indicted the governor’s closest aide, Joseph Percoco, in the fall as part of a wide-ranging bribery and bid-rigging investigation. Mr. Cuomo has repeatedly described himself as shocked and heartbroken by Mr. Percoco’s alleged actions, but a conviction could reflect badly on the governor, as could Albany’s reputation for corrupt politics.