I use a credit card for online payment only. I don't like the idea of a debit card stored online because it is a direct link to my bank account. A credit card gives me a bit more protection and I make sure I pay it off every month.
FYI I got a "prepaid" card for that - it's a card I use for online payments only - it has no credit and it doesn't have access to my account funds - I need to manually transfer money when I want to use it to pay for something. If it ever gets stolen online there's nothing on it unless I'm right about to purchase something and I always put in the exact ammount I'm about to pay so I can't get overcharged or some other silliness.