U.S. regulators proposed a record fine for alleged unauthorized drone flights as they intensify efforts to crack down on unmanned aircraft that they say are endangering U.S. skies.

The Federal Aviation Administration Tuesday proposed to fine SkyPan International Inc., a small Chicago-based aerial-photography firm, $1.9 million in connection with allegedly flying drones in restricted airspace above New York and Chicago 65 times from March 2012 through December 2014, including 43 flights near New York’s large airports. Previously,...