We had a good time at Big Bend, the one part of the park that we didn't make it too was that SE corner that you are talking about. But there is nothing to keep anyone from wading across the river. And down in the state Big Bend Ranch Park the river vanishes every few miles so they won't even have to get their feet wet.
I blamed the Feds for not building a border fence but recently I heard that the Texas state government told the Feds that they didn't want a fence. My wife said she saw a sign saying there was an $8000 fine for unauthorized border crossing, I missed it. We more of less drove right along the river for 60 miles in the NP (river road) and another 40 or 50 in the State park. Hardly saw another person the whole time.