Devils Garden Campground (elevation 5,355 feet, open year-round with water, $20) is near the end of the 18-mile scenic drive [1]. It's an excellent place to camp, with some sites tucked under rock formations and others with great views, but it's extremely popular.
The well-organized traveler will plan accordingly and reserve a site in advance for nights March-October. Reservations (www.recreation.gov [2], $9 booking fee in addition to camping fee) must be made no less than four days and no more than 240 days in advance. Not all sites are reservable; 24 remain open on a first-come, first-served basis. During the busy spring, summer, and fall seasons, campers without reservations should plan to queue up at the front door of the visitor center [3] at 7:30 a.m. sharp for a chance to fill a campsite.
If you aren't able to score a coveted Arches National Park [4] campsite, all is not lost. There are many BLM campsites within an easy drive of the park. Try the string of primitive BLM campgrounds on Highway 313, just west of U.S. 191 and on the way to Canyonlands National Park's Island in the Sky District [5]. Another cluster of BLM campgrounds is along the Colorado River on Highway 128, which heads northeast from U.S. 191 at the north end of Moab [6].
Links:
[1] http://www.moon.com/destinations/utah/canyonlands/arches-national-park/scenic-drive
[2] http://www.recreation.gov
[3] http://www.moon.com/destinations/utah/canyonlands/arches-national-park/visitor-center
[4] http://www.moon.com/destinations/utah/canyonlands/arches-national-park
[5] http://www.moon.com/destinations/utah/canyonlands/canyonlands-national-park/island-the-sky-district
[6] http://www.moon.com/destinations/utah/canyonlands/moab