As a previous Employer of
Enterprise and
Budget, I would never use a debit card for a car rental. Since a debit card is tied directly to your checking account, it can be an accident waiting to happen. I have seen MANY customers who no matter how you explain to them that there will be a hold on funds in their checking account, they don't seem to understand. And then once the rental is over, it can take days for the hold to come off. The delay on the hold coming off is not the rental company's fault but the creditors procedures that are in the way.
I would definitely use a credit card. If you don't carry a credit card when you traveler it may be a good idea to get one. I have a small $500 credit limit on one just for car rentals and ordering items on line.
Debit cards can be very convenient for your use, but it is also very convenient for others to steal your numbers and use as well.