I agree with Jeff about when to go. One other thing.....don't drive over.....take a taxi or walk.
There used to be some really neat crossings such as Boquillas in Big Bend and Lajitas, which is just up river out of the park. They would take you across the river in a jon boat, then give you a burro or pickup into town to eat, however following 9-11, those crossings were closed.
Larry