In a perspective this can be seen as a frame challenge.
An imprisoned mage also represents opportunities. Just like any prisoner, you can use them as labour. Their magic can be a great boon for the people who imprisoned the mage in many cases. If this is the case, the deterioration can be halted or reduced, as the mage sometimes can use the arms and hands.
This isn't necessarily a common practice. Forcing prisoners to work needs to be advantageous. They can do tough jobs for extended times, while a lesser amount of people equipped with weapons make sure they do good work. Mages will be more difficult to control. They need to be guarded by someone vigilant and consistently more powerful. If they have artifacts to help them it would be great, but if the power comes only from the mage this can be difficult.
Your mage might still be restricted, or forced to work in such a way that little power is left for revolting (like dehydration, sickness, hunger or magical exhaustion). Or maybe the mage is someone with a weak spot, so if the mage revolts the family is slaughtered. That way the mage was able to move and maybe even practice magic, have had a a gruelling life in prison and a plausible reason why she/he didn’t escape.