I've been using debit cards since they came out (perfect credit BTW, so certainly not a case of "marginal credit" as some other poster inanely commented); Also have several CC's that get paid off after each purchase.
I've had my debit and CC numbers skimmed / scammed multiple times as I travel a lot and you'd be surprised how easily someone can get your card info at a gas station.
In any event, I've actually had more proactive protection and action from the debit card than the CCs, I'll get fraud alerts very quickly and its the same process from both when they catch it and correct, send a new card, etc.
To your point regarding overdraft fees, it's never happened to me in over 25 years, they automatically refund the fraudulent charges and I've never paid an overdraft fee of any kind.
Perhaps it all boils down to which bank you use & their policies, but my wife and I consistently use our debit cards without fear.
Sure, for big purchases, we use our CCs for the added insurances they provide along with the points / miles that we rack up, but for day-to-day items, I much prefer the convenience of tapping / swiping a debit card than carrying around stacks of cash. If my wallet gets swiped, I touch a button and my cards are frozen, but say goodbye to the cash....
For an online purchase of an implement, CC all day long