The governor said health officials did not believe the woman was contagious when she was flying from Iran to New York, nor when she took a private car home from the airport.

“But out of an abundance of caution, we’ll be contacting people who were on the flight with her from Iran to New York and the driver for that car service,” Mr. Cuomo said, though he did not have the specific details of which flight she took.

Mr. Cuomo also said the state would institute new cleaning protocols in crowded public places, including schools and busses. “If it smells like bleach when you get on a bus or when a child goes to school, it is not bad cologne,” he said. “It is bleach.”

He said that the public health system was focused on reducing the spread of the virus, but suggested that eliminating it in New York was likely to prove impossible in the short term.

“That’s all this is about, reducing the spread, not eliminating the spread,” Mr. Cuomo said.

For weeks, the city has been bracing for the virus’s arrival, as it spread across much of the world.

By this weekend, several stores in the city had run out of sanitary supplies such as hand sanitizer, masks and antibacterial wipes. Some people had tried three or four stores in search of hand sanitizer, before admitting defeat. Boxes of surgical masks that sold last month for $20 were going for $75 at one grocery store in Chinatown in Manhattan.

At Costco in Brooklyn, people were stocking up on bottled water, Clorox wipes, cold medicine and cans of food, anxious for the days and weeks ahead.