I guess if there were kidnappings, murders, or robberies against visitors to Big Bend, it would make the news in southern TX at least and might even cause the park service to issue an advisory or something. I haven't researched to see if there are any such signs of problems and significant danger, but I haven't heard any stories so I'm going to go out on a limb and assume that it's probably pretty safe.
That said, if I were to visit Big Bend personally I would remain alert, lock my RV when not around it, and carry my usual self-defense tool.