When Pam Mundus and a friend landed in Milan for a weeklong vacation on Feb. 23, uniformed airport workers in face masks, attempting to contain the spread of the new coronavirus, took their temperatures.

But when Ms. Mundus returned home on a direct flight from Rome to New York’s John F. Kennedy Airport on March 1, no one questioned her about her time in Italy or whether she might have been exposed to the illness. At that stage, the State Department had already urged Americans to reconsider travel to Italy; there were at least 1,500 cases there and 34 deaths related to the coronavirus by then.

“The only question we were asked was, ‘Have you been to China?’” Ms. Mundus, 62, said in a phone interview from her home on eastern Long Island, where she has been in self-quarantine since her return.

President Trump has claimed credit for slowing the spread of Covid-19 in the United States by imposing a ban in late January on some travelers who had recently been to China. His administration has since barred entry to travelers who have been in Iran, nearly 30 European countries and the United Kingdom.