NASA’s plan to routinely ferry astronauts into orbit using private spacecraft—initially slated to start last year—has now slipped until at least the spring of 2019 and unresolved hazards threaten further delays.

New questions about the high-profile program, known as commercial crew transportation, emerged Thursday, with the National Aeronautics and Space Administration’s top outside safety panel raising red flags about persistent dangers. Under the program, NASA would rely on capsules developed and operated on a commercial basis by Boeing Co. BA, -3.81% and Tesla CEO TSLA, +4.42% Elon Musk’s SpaceX.