I get asked to film theater, standup, and improv shows fairly often, and I've seen plenty of videos of the same, and I'm going to share a tip with you that will improve your theater (etc.) videos by 45%*. We're going to learn to use manual exposure.
When you leave your camera on automatic exposure, it's dealing with averages and it sees the vast expanse of the black curtains and jacks up the overall exposure to try and get some detail in those areas -- leaving the human face, the part of the image you actually care about, completely blown out. It's even worse if everyone on stage is wearing dark clothing. I've seen videos where everyone's heads looked like they were on fire.
So what you want to do is find the manual exposure setting on your video camera. On this Sony Handycam I've got here, for example, you go into FN on the touchscreen, then EXPOSURE, select MANUAL and use the - and + buttons to set the exposure. Press OK and then X when you've got it set to step out of the menus. Just dial down the exposure until you can see detail in the face. It's subjective -- too far down and the whole image will be pretty dim.
Ideally, you can get into the space early enough and get cooperation from the tech people to take out the house lights, turn on the lights the way they'll be during the show, and position someone mid-stage to set your exposure (that's what I was doing here with the aid of the delightful Paul O'Toole). Or you might need to wait until the host is on stage. Or just dial it in pretty quickly during the start of the show.
* A fake statistic.