About the only downsides to such a magnificent park are:
1. Proximity to Mexico (although in and of itself isn't a bad thing, but there's a reason why they have a permanently manned border control post right outside the park)
2. Inability to carry firearms into the park (for self defense against point 1, not for hunting)
3. Inability to bring a drone into the park for some amazing aerial shots.
Though I didn't actually compile this before heading to Big Bend, here's what my incomplete "List of Things to be Concerned About" might have been, in order of most concerning to least concerning:
1. Am I going to get a flat tire?
2. Are we prepared for large temperature swings?
3. Are we going to run out of water?
4. Are we going to run out of food?
5. Are we going to run out of gas?
6. Are we going to get lost?
.
.
.
44. Are we going to encounter a rattlesnake?
59. Are we going to encounter a bear?
83. Are we going to encounter a mountain lion?
.
.
.
125. Are we going to have fun?
126. Is my 4Runner going to fail me?
127. Are we going to encounter a hostile illegal immigrant?
128. Do I need a firearm to defend myself against said immigrant?
Obviously, I'm poking fun, but your #1 and #2 downsides would be the least of my concerns when visiting Big Bend. Just my opinion. Illegal crossings in the park are possible, of course, but I can't imagine a worse place to cross. Among other things, they're contending with hundreds of miles of harsh desert and little to no resources.