Even top officials at the department acknowledge the task of sealing the United States from the virus is impossible.

“This has never been from Day 1 intended to be a hermetically sealed process,” said Kenneth T. Cuccinelli II, the acting deputy secretary of homeland security. “We are trying to reduce and delay the biggest peak in the virus wave hitting on the United States of America. And all of these steps reduce and delay. They do not stop of the virus.”

“Viruses do not care about boundaries,” he said.

Maggie McDow, 46, said she had swollen glands when she landed at Dulles International Airport outside Washington, one of the airports where travelers who have recently been in China or Iran are being funneled. Ms. McDow had flown from London, but before that, she had stopped in South Korea, a coronavirus hot spot.

Ms. McDow said she heard plenty of instructions from airport officials for those feeling ill in airports abroad. But when she reached the United States, she said, “There was none of that.” A customs officer stamped her passport, and she was on her way.