I would stay at each of the parks and take the shuttles to get around. There are campgrounds in each park if you get there early (before noon) that are very reasonable. We spend at least two to three days at each park for sightseeing. We used to hike all the trails but not so much anymore. To set up a base camp between the two parks would mean you would be doing a ton of driving back and forth and not seeing much. That's what we do and we've been there like three times. Don't forget to visit the arches, and all the other places out there.
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