This movie, of course, was Zero Hour! from 1957, starring Dana Andrews, Linda Darnell, Sterling Hayden and Elroy "Crazylegs" Hirsch.
A large portion of this movie would later be parodied in the 1980 movie Airplane!. TCM showed both movies back to back and although Zero Hour! was a drama, it was quite amusing seeing just how much of the movie was copied scene-for-scene and even line-for-line in Airplane!. Lines like "I guess I picked the wrong week to quit smoking", "I may bend your precious airplane, but I'm bringing it down" and "Flying a plane is no different than riding a bicycle." I laughed when they took the young boy up to the cockpit and the pilot hands him a toy DC-4, puts his arm around the boy and asks, "You ever been in a cockpit before?". Even the scene with the woman who goes crazy and ends up getting slapped by a line of people makes an appearance.
If you choose to watch Zero Hour!, I'd highly recommend making it a double feature with Airplane! first.