Credit cards have better fraud protection. It is actually suggested that you "never" use a debit card online, as they can clean out your checking account.
Both debit and credit cards usually have protection against fraudulent charges. Sometimes it takes months for these to be resolved though. My debit card is tied to my checking account. I depend on this for my basic needs. Being without the money for a couple of months would be a disaster. It may also cause me to bounce checks while the bank is trying to figure it out.
I always use my credit card for online purchases. If you pay your balance in full at the end of the month it doesn't cost any extra.
Credit card is the better way to go. If there's a problem, it's just a little of your line of credit tied up. Use a debit card and it's actual money out of your bank account. It's also easier to get credit cards to process disputes quickly.
Credit card, with a debit card if someone steals your information they have access to your bank account.
credit card definitely