Where I live, many businesses have key-coded locks on their bathroom doors. A customer/employee punches a 4 number code on a small pad attached to the door handle. A person only gets the code if they purchase an item as it's printed on the receipt. The door self locks and closes automatically.
I am not sure how expensive it would be to place such a lock on the laundry door and rig the door to self close, but you may want to get an estimate. Tenants would get either their own code or it would just be one code distributed among all.