As others have suggested, you're very unlikely to have problems with illegals but do exercise common sense when hiking or driving the backroads. Other than the period between Thanksgiving and New Year's and Spring Break in early to mid March, Big Bend is one of the least visited National Parks. And that's a good thing!
Just to add a little spice to your visit, this
1932 murder is detailed on the Park's website. Juan's grave is located on The Old Ore Road and well worth a visit. Also,
Death in Big Bend by Laurence Parent is interesting reading. And
A Murder in Terlingua was a multi-part TV special a few years back.
Seriously, we're from NC but Big Bend is our favorite National Park and we've returned many times since spending part of our honeymoon there in 78. In fact, we'll be there for a few days in late June.