I am hyper-mobile due to Elhers-Danlos type III. I just need to make sure that my knees and shoulders are protected. I found that losing weight and gaining muscle/strength actually makes my joints MORE stable, less prone to injury, and less painful.
As was mentioned, if you have a specific diagnosis, talk with your doctor not only to protect your joints, but your heart as well (very often, hyper-mobility syndromes have issues related to the body's connective tissue, including those found in the heart).
Trying like crazy not to fit into my genes.