I completely agree with @littleadv in favor of using the credit card and dispute resolution process, but I believe there are more important details here related to consumer protection.
Since 1968, US citizens are protected from credit card fraud, limiting the out-of-pocket loss to $50 if your card is lost, stolen, or otherwise used without your permission. That means the bank can't make you pay more than $50 if you report unauthorized activity--and, nicely, many credit cards these days go ahead and waive the $50 too, so you might not have to pay anything (other than the necessary time and phone calls).
Of course, many banks offer a $50 cap or no fees at all for fraudulent charges--my bank once happily resolved some bad charges for me at no loss to me--but banks are under no obligation to shield debit card customers from fraud. If you read the fine print on your debit card account agreement you may find some vague promises to resolve your dispute, but probably nothing saying you cannot be held liable (the bank is not going to lose money on you if they are unable to reverse the charges!).
Now a personal story: I once had my credit card used to buy $3,000 in stereo equipment, at a store I had never heard of in a state I have never visited. The bank notified me of the surprising charges, and I was immediately able to begin the fraud report--but it took months of calls before the case was accepted and the charges reversed. So, yes, there was no money out of my pocket, but I was completely unable to use the credit card, and every month they kept on piling on more finance fees and late-payment charges and such, and I would have to call them again and explain again that the charges were disputed... Finally, after about 8 months in total, they accepted the fraud report and reversed all the charges.
Lastly, I want to mention one more important tool for preventing or limiting loss from online purchases: "disposable", one-time-use credit card numbers. At least a few credit card providers (Citibank, Bank of America, Discover) offer you the option, on their websites, to generate a credit card number that charges your account, but under the limits you specify, including a maximum amount and expiration date. With one of these disposable numbers, you can pay for a single purchase and be confident that, even if the number were stolen in-transit or the merchant a fraud, they don't have your actual credit card number, and they can never charge you again.
I have not yet seen this option for debit card customers, but there must be some banks that offer it, since it saves them a lot of time and trouble in pursuing defrauders.
So, in short:
If you pay with a credit card number you will not ever have to pay more than $50 for fraudulent charges.
Even better, you may be able to use a disposable/one-time-use credit card number to further limit the chances that your credit is misused.
Here's to happy--and safe--consumering!