WASHINGTON—The Trump administration began enforcing its temporary ban on refugees and on visitors from six Muslim-majority nations on Thursday evening, exempting only those travelers with close family or business ties in the U.S., officials said.

The guidelines are a narrow reading of a Supreme Court ruling on Monday that required the U.S. to continue to admit people who have “bona fide” relationships—with either people or institutions—into the country. Under the new rules, for instance, a sister or a husband of an American...