Customer was afraid she would loose her debit card so she took a picture of the front and back of her debit card and saved it on her phone. As you might guess the phone was stolen and now we are getting pay pal charges on the account.

Customer has reported the charges as unauthorized. Does the customer ever have any responsibility ion a situation like this?

I don't think so. It sounds like the equivalent of putting the PIN on the back of the card, which is specifically called out in the commentary as not shifting liability to the customer.

Correct. Customer negligence means nothing to Reg E liability. Generally the bank will win chargebacks for online purchases so you may be able to limit your losses. I’d also limit my losses by not giving this customer another card.