There is a border crossing in the park. It is basically a dirt road. If you drive your $100k car down there it could disappear over the border very quickly. You may encounter Mexicans selling geodes most likely. They might be watching for potential marks as well. There may also be Mexicans on foot that want a ride up the road. They may be pushy and mob your car, sometimes in large groups.

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.