In the begining of September we visited Paris for one week.
A great website to look for apartments in Paris is www.parisattitude.com. We got a small one bedroom apartment with sofa bed in the living/kitchen/dining area for less that 500euros a week in the Moulin Rouge area. They will send you an email letting you know if the apartment is available, then you will be required to get in contact with the owner and send them signed copies of the rental agreement and 100euros to half the cost of apartment for a deposit. Many of them accept Western Union (most expensive), Paypal (best) and Bank transfer (expensive). When you arrive you pay the owner the remaining amount + some will ask for the same amount again as a safety deposit that you will get back when you leave (this is to cover any damage that may occur). We have found that many look at our children and decide that they are old enough to not destroy anything and forgo asking us for this deposit.
The Moulin Rouge area is classed as "red light" so many of you are thinking - oh no I would never stay there with my children - Well trust me I thought the same thing when we arrived at the apartment and I could see all of the shops and "dance" halls lit up (we arrived at midnight). But the next morning I looked out of the window and it was like any other city, families out for walks, business people on the way to work, people walking their dogs and stores opening up for the day. The thing is when you are travelling with children you are not likely to go out a night except for dinner and normally with children that is quite early, so you don't really see anything that goes on at night. Also in Europe they quite often use double glazed windows for the apartments in built up areas, so when inside with the windows closed you don't hear a thing.
Back to our apartment, the owner met us at the door when we arrived and showed us in to a very secure apartment building. He showed us how to turn the key to enter and where all the household equipment was stored (extra blankets, iron etc...), how to use the washing machine and cooktop. He gave us a map of the city and showed us all the areas nearby that are "classic Parisien" and told us we must see this and that. He was very informative and extremely helpful.
If you are travelling with a pram/pushcar or buggy, please be aware that in Europe you will not find many apartments that have lifts, you will more than likely get an apartment that has between 3 to 7 flights of stairs to climb, so keep this in mind. Always ask if they are on the ground floor or 1st floor (you should be fine up to the 2nd floor).