Is a credit card number really valid for a deposit on a service?

I own a small business which often takes credit card numbers over the phone as a deposit. We recently had a customer no show, and she contested through her credit card company when we charged her (she claims it wasn't explained when she gave us her #). The credit card company took her side and said... I get a credit card? How do you choose the best credit card? What credit card company offers the best deals on Business Credit Card? What types of credit cards pays the customer for using the card