Joe, we felt perfectly safe and hiked within 20 feet or so of Mexican land with the Rio Grande inbetween. We never saw anything, no one trying to cross, no deflated tires, no disguarded clothing or plastic bags that are used to transport dry clothes.
You will pass through Border Patrol checks going out of the area, we passed them leaving Tombstone Arizona, Las Cruces New Mexico, etc. We never had to stop more than to say, yes, we are US citizens and nope, no one is in that 5er.
I have just about finished blogging about our experience, go have a look, here is a direct line to one of the hikes, Santa Elena, thanks to Steve R's suggestion!
Santa Elena hike.
If you snoop around you will find several more posts about Big Bend.
It might start to get warm down there in April, but, I hear that January, February and March the campgrounds are quite full all around the Big Bend area.
I cannot say you will not feel uncomfy, but, we certainly did not, and I refuse to go into Mexico at border crossings anymore, so, take it for what it is worth with that in mind. And, if you feel uncomfy, then, pick up and get out! Safety first.
But, our stay was very nice, very quiet, not many people in the park. Did I say it was quiet?? WOW, was it quiet!