We found a great BLM campsite (free) 20 minutes from the Caverns (more infos and GPS coordinates here).

This place is just out of this world magical... we had visited a good chunk of it 3 years ago, but hadn't done the back part. The entrance is free with a National Parks pass. As amazing as it is to enter the caverns through the natural entrance (walk down into the caverns), it is quite the hike to do the whole thing (adults can do it, but it's long for younger children). Here's our advice: arrive early and take the elevator down to the big room and walk all the back part of the caverns. You will have the place pretty much to yourself since most people are entering through the natural entrance at that time of day. Then, either walk out through the natural entrance or take the elevator back up. Walking into the caverns is a unique experience that is most enjoyed when you are not stuck in a crowd.