Do you know how annoying it is when there are no paper towels?? No hand dryer, no paper towels, you just have squat (pun intended). So you use the restroom, wash your hands and there you are with wet hands.
What I would recommend in this situation is suing the establishment that left you paper toweless. By having wet hands and opening the door you open yourself to a whole world of germs that are activated by the water that is on your hands. The water helps to scrub the germs right off the door handle and right onto your hands.
By suing these establishments you can send them a clear message that if they want to endanger public health by not providing paper towels in their restrooms then they can pay you an advance against the future health bills that you are going to have to pay because of their oversight.
Your other options involve kindly asking a customer service rep, using toilet paper, or making due with wet hands. Those are not very fun though. Suing is a blast!