The Rio Grande River runs along the Park's southern border, serving as the international border with Mexico, as well. To raft the river with your own equipment, you need to get a river float permit - free - at one of the Ranger Stations. You can also buy a printed river guide from here too. There are no equipment rentals in the Park though there are several outfitters just outside, where you can arrange for rentals and/or shuttle services or simply get a river trip with guide and all.
River flows are heaviest in the July-October thunderstorm season and not at the end of spring since much of the upriver water is used for agricultural purposes. The water here mostly come out of the Rio Conchos in Mexico and is more considerable than the little trickle you see up in El Paso. There are three glorious canyons to float through in the Park - Santa Elena, Boquillas and Mariscal. Downstream from the Park, the River continues in a protected state as a Wild and Scenic River. It is possible with an outfitter or by yourself, to spend a week or more floating through this remote wilderness.
http://www.nps.gov/bibe/float.htm for river guides and rentals
Texas River Expeditions, Study Butte.....371-2633 or 800-839-7238 http://www.texasriver.com/
http://www.nps.gov/bibe/riversb.htm for nps floating information
At the end of the Ross Maxwell Scenic Drive you can decide to backtrack or you can take Old Maverick Road which leads you to a spot near the West Entrance to the park and the towns of Terlingua and Study Butte. Be advised though that the road is not paved. Check on road conditions before taking this road.
There are three paved roads within the park and a number of unpaved roads. There is no public transportation offered within the park, so you will have to supply your own transportation, both for getting to the park, and getting around within the park where you can see views such as the Cerro Castellan with its volcanic layers along the Ross Maxwell Scenic Drive, and access trail heads.
For filling up your gas tanks there are two gasoline stations in Big Bend. One is at the Rio Grand Village on the southeastern side of the park, and the other is at Panther Junction, the most centrally located Visitor Center and Park Headquarters. Both stations have gas, diesel, and can do a few minor repairs. The Rio Grande station also has propane.
Both to go to Big Bend National Park, and to visit it, you need a car. There is no public transportation. RV and big buses are not advised to drive up the Chisos Basin (too twisty tiny road). A 4-wheel drive is advised for most dirt roads.
Take gas at Alpine (100 miles Northwest) or Marathon (50 miles North) or Study Butte/Terlingua (a few miles West), as the gas stations in the park are very expensive. They are located at Rio Grande Village and Panther Junction.
Unless you can find a tumbleweed heading your direction, the only way to get here is by car. No airports, no train stations...I suppose if you REALLY wanted to you could float here from El Paso....
Do bring a high clearance vehicle if you can though, there are some great driving paths in the park that you just couldn't do with a Ford escort...