When somebody steals your debit or credit card number, you're not responsible for the charges as long as you notify your bank promptly.
Does it matter whether or not the card was a debit card or a credit card? Certainly. Credit cards are more consumer-friendly when it comes to fraud protection (although they're less friendly if you carry a balance and pay interest). Just notify your credit card issuer of unauthorized use, and you're not responsible for more than $50 - you might not even have to cover that much depending on the circumstances.
With a debit card, time is of the essence. You need to notify the bank within two days of a bad transaction to limit your losses to $50, and after 60 days you might be out of luck completely. What's more, that money is missing from your checking account, and that might cause some inconveniences (unless you keep a lot more in checking than you'll ever need). Your bank will replace the funds, but as Claes Bell explains, the process is probably slower than you'd like, and "banks aren't exactly falling all over themselves to make sure you get your money in time."
Given all of that, think twice before using your debit card. If the card information is stolen and money disappears from your account, how easy is it for you to cover your bills? If the answer is "not very," then use a credit card for everyday purchases and online shopping. Just be sure to pay off the entire balance every month.