Mr. Cotton, who oversees the region’s airports and other transportation infrastructure and had been visiting Port Authority facilities, was being quarantined, Mr. Cuomo said. His team was also being tested for the virus.

Mr. Cotton is one of the most prominent public officials in the United States to have contracted the virus.

The governor also said that the state would soon require that if a student in New York tested positive for the virus, their school would be closed for an initial 24 hours while health officials assessed the situation.

On Monday evening, the New York Fire Department announced that an Emergency Medical Service worker had tested positive for the virus after coming into contact with a person who had traveled overseas.

The worker was asymptomatic, the department said in a statement, but had worked three shifts at a firehouse and had treated 11 patients in the time before testing positive. All of those patients had been notified of the worker’s positive test result by the Department of Health, the statement said. The worker’s colleagues had been directed to self-quarantine.

In response to concerns over availability and price gouging, Mr. Cuomo said the state had asked Corcraft, a state entity that uses prison labor to manufacture products, to make 100,000 gallons of hand sanitizer a week.

The state planned to distribute the product for free in the communities and government agencies most impacted, including the Metropolitan Transportation Authority.