From my experience with individual, not multiple, swollen joints (ankle & elbow tendinitis, bursitis-shoulder), I would suggest strength training to support the joints and limited repetitions or impact. I think I've heard that water exercise is beneficial, too.
When my shoulder was at it's worst, I could only do lower body exercise, but as it improved after a cortisone injection, I would take an NSAID, slap on a salon-pas patch, and then work out, but with the knowledge that I had a limited number of times I could move the joint. So I was able to rock climb (indoor, with ropes, no jumping, calculated moves) for like four times longer than I could take a dance class (arms waving in the air) and a yoga class with tons of "planks" set me back quite a while at one point.