We have family land that is held as a corporation. Each family member is entitled to a lot to build a house and the lot is transferd by quitclaim deed. If someone gets a mortgage and then the property is foreclosed on, is the corporation in anyway responsible for the mortgage of a family member?

As long as the lot was transferred by deed and the deed was recorded, then the corporation should have no responsibility for the mortgage that the family member places on the lot after the transfer. The only way the corporation would be responsible is if the corporation co-signed or guaranteed the loan.