In reply to djwilse:
We stayed here a few years ago -
http://www.campingdelile.fr/welcome.htm
It isn't in the middle of the Jura but is reasonably well placed. Run by a very nice Swiss couple and busy with Swiss campers. Big pitches and good local amenities. The near by main road is a bit noisy though.
Overall you will struggle to find any swimming in the rivers in this area since the water is pretty cold. We took an inflatable canoe and that was great. There are plenty of recreation lakes in the area though. The cragging is excellent with loads of crags around Baumes Les Dammes.
Didn't look for canyoning or VFs but I am pretty sure both will be available. Besancon is a great town to explore as well with an amazing castle with a zoo in it.
We drove to the big mountains to get some uphill road biking in and there may well be camp sites better situated for this in the Jura area.
Alan