They might check the Net for prices and products. But for actual purchasing, they visit the store or pick up the phone. Card issuers, hoping to calm these fears, are always on the lookout for new technologies that might make you feel safe. The latest is a system for creating disposable credit card numbers. There's nothing in the merchant's data bank for hackers to steal.
Shoppers have two security concerns.
First, they worry that their credit-card numbers will be stolen. As a practical matter, that's a minor issue. The second, far greater problem is identity theft. You'll be dunned and sued. It can take a year or more to straighten out the mess.