In a previous job I held, there were eight of us in my department. All of us were out of town half the month. We all had cards that we only used at hotels. In a year's time, four of us had fraudulant charges on these cards after using them at hotels.
Now, I'm not knocking hotels here, the same thing can happen at any merchant. However, that's no reason to give them your private info. By the way, it is a driver's license. There is no law requiring you to have identification everywhere you go. You don't have to show identity papers at Walmart or anywhere else. Your CC, along with your signature, is your ID as far as your transaction with the merchant is concerned. This is per the agreement between card and merchant.