WASHINGTON—The House overwhelmingly approved legislation Tuesday to limit certain travel privileges granted to citizens of 38 friendly foreign countries, the first step in what lawmakers expect will be a larger response to an evolving terrorist threat.

The terror attacks in San Bernardino, Calif., and Paris have prompted a new emphasis on security but left lawmakers struggling to determine the appropriate response. The strikes reach into so many different policy areas—travel, guns, technology, mental health, immigration and...