We visited BBNP for a few nights and absolutely loved Chisos Basin campground. The site (41) was perfect. We had unobstructed views of the Basin (see photo), far enough way from the restrooms and water spigot that we didn't get a lot of through traffic. There aren't many trees in the campground, so you can see your neighbors, but the sites are far enough apart that you're not completely on top of one another. There are numerous hiking spots inside the Basin, or you can venture further into the park to visit either of the canyons or numerous desert hikes. At first we didn't like the fact that there are no fires allowed in the Chisos campground, but once night fell and we were able to stare at the Milky Way for hours and hours, sipping our wine in a quiet campground, we completely forgot about the lack of campfire. LOVED BBNP.

Edit: you likely won't be able to stake your tent; recommend bringing sand bags or expect that you'll have to find rocks nearby to weigh your tent corners and tie your guy lines.