“The New York concern is that these are people who are coming into New York to work,” he said. “It would have been easy but irresponsible for me to say it’s not my problem.”

It became his problem, at least in part, when the Port Authority of New York and New Jersey became the sponsor of the Gateway project. Mr. Cuomo controls the Port Authority, along with his counterpart in New Jersey, Gov. Philip D. Murphy, who like Cuomo is a Democrat.

Officials in the two states have been pushing for a new train tunnel across the Hudson for years. Work had begun on a project that preceded Gateway, but in 2010, Chris Christie, then the Republican governor of New Jersey, canceled it.

The need for a new passageway grew much more dire six years ago when Hurricane Sandy flooded the rail tunnels with “a toxic stew of Sandy salt water,” said Rick Cotton, executive director of the Port Authority. Since then, he added, the residual salt has been eating away at the concrete-and-steel walls of the tunnels and corroding the high-voltage cables that provide power through the tubes to Penn Station.