Border Patrol
When I was in the Cave Creek Canyon area, there was a noticeable Border Patrol presence. In fact, they had made an arrest and were bringing someone out on a trail.
I had no problems while I was there, but seeing the arrest did prompt me to stay in a developed campground, instead of alone in the backcountry. It turned out fine, because I stayed in Sunny Flat campground, which has some great views of the canyon rock. I recommend this area if you're going to be in the Chiricahuas, though it's not close to Cochise Stronghold.