Unless you have a 'debit only' card, I don't think it matters. My debit card can also be used as a regular credit card, but not sure if that is true for all cards. When traveling and using an out of town gas station (for example), when the screen asks debit/credit, I always select 'credit' and then it asks for my zip code instead of the 'pin #' used for debit purchases. I feel there is less risk giving out a zip code with a merchant I am not familiar with that covers a large area vs. your very specific pin # which could allow for fraud. As someone else mentioned above, the one time a suspicious purchase appeared, the card company immediately deleted that charge and there was no liability on my part. In general, I don't feel there is any more/less risk associated with the type of card you use.
I saw something recently at the airport that there can be issues using a debit card for rental car reservations, but I don't recall all of the details.