It's scary, stupid and they're just not going to do it. That already leaves you at a starting point where you don't have to be all that worried. Which doesn't mean don't worry at all. There are some important measures you should take before keying in that long list of numbers and expiration date. First, be sure to use a credit card, not a debit card, when shopping online.
Debit cards are much more risky than are conventional credit cards.
If someone gets your debit card number, they can drain all your bank accounts. You are not guaranteed that protection if you use your debit card. Next, credit card in hand, be sure you shop only on secure connections. That's where sites use encryption, which basically scrambles the information as it goes careening around the Internet. In order to confirm the site is safe, look for that little padlock on the bottom of your browser.