We've used parisaddress.com a couple of years ago to rent a studio for a week and were very happy. The apartment was clean and the dudes from the agency were friendly. Someone usually meets you in front of the apartment when you arrive and comes by to check you out, pick up the keys and call you a cab if needed. Whomever owned the apartment was even nice enough to leave his printer for us to use so we could print out our boarding passes when flying back. Having your own kitchen, fridge,etc, is pretty cool and you feel less like a tourist if you are not staying in a hotel. Remember to say Bonjour to the neighbors.

There are a few other agencies that offer short term rentals if you Google. I think I paid a deposit using a credit card and the balance in cash in person though terms may have changed by now. Be aware that some listings may want extra cleaning fees or breakage/damage deposits on top of the rental fee.