WASHINGTON—The Trump administration offered guidance to officials to maintain a measured response to antigovernment protests in Hong Kong over fears that any public statements favoring demonstrators would derail U.S. efforts to get a trade deal with China.

As hundreds of thousands of protesters began taking to the streets of Hong Kong starting in June, administration officials were working to revive their collapsed trade talks with Beijing. Hong Kong is facing its worst political turmoil since the former British colony was...