A stage crew was a group of people (stagehands) responsible for assembling and tearing down a play set.
According to an anything but canon account, while celebrating the TOS crew's fiftieth anniversary party, Montgomery Scott was angered by something that D'Vana Tendi told him. She told them "You set the stage for modern Star Trek cartoons." He took it to mean that she viewed them as stage hands rather than actors. He even told her that what she said sounded like an insult. (VST: "Walk, Don't Run")
In 2369, Beverly Crusher and the stage crew for Frame of Mind planned to take down the set the day after the last performance. Following William T. Riker's experience with the Tilonians, he decided to disassemble it himself that very night, believing he wouldn't be able to sleep if it was still up. (TNG: "Frame of Mind")