I'm in my 20s female traveler and would probably be traveling alone, most likely sleeping in the back of my SUV- which I've done all over the country and generally in sketchy areas I'll get a hotel room or rent a real campsite.
In terms of safety, it's not a place where anyone "looking for action" is going to hang out. It's a long way from anywhere and even at peak capacity mostly empty. That's my take on most wilderness areas in the US and the only locals at Big Bend are in Boquillas and dependent on a good tourist experience.
All with the caveat that I'm an old man not a young female.
Anyway, the biggest dangers are the usual suspects: exposure, dehydration, and falls. It's remote and dry with temperature extremes. The average worst event is a flat tire from taking Old Maverick Road without truck tires on your vehicle.
If you stick to camping, motel, or car camping in Chisos basin then you're probably good. There are wild havalinas which are mini-pigs that will root around at night, not dangerous.