I use by bank-linked card as a credit card, so it works like debit even though it's credit. So there's no need to pay it off.
This is a bad idea since most credit cards are protected against fraud but many checking accounts are not. (At least not to the same extent.)
Up above, the other poster who keeps a small credit limit to protect against fraud is being a bit overprotective. If someone in Russia buys 100 TVs with his credit card, he'll probably be able to get that fixed, no matter what his credit limit is.
But in your case, you'd have a MUCH harder time fixing that if someone steals your debit card. Get a credit card and keep good track of what you're spending and just pay it off each month. It's a lot safer.
EDIT: I see Babaganoosh has beat me to it with some links. All good advice.