That is a heart-breaking problem in a quilt! You have my sympathy. Wish I could magically fix it for you, but I haven't yet perfected that quilting technique.
The fabric appears to be ripping from the quilting, so I'm going to take a wild guess and say it's dry-rotted. My opinion - as someone who has never dealt with this problem - is that the backing should be removed. The dry rot will continue to spread throughout the backing.
Putting another back over it might work, but might not because as the fabric deteriorates it is likely to break away and ball up under the new backing. It also adds another set of stitching for the fabric to tear away from.
If I were in your situation, I would remove the back simply so I never have to deal with it again.