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If you don't get a recommendation, I have a well-traveled friend who makes excellent use of concierges at hotels where she is not a guest. She simply tips them well, and her wish is their command.

You might also enquire at one of the American churches in Paris, the Episcopal cathedral, for example.

Thanks Ackislander - good thought! Unless anyone has a recommendation.

I'm sorry I don't have a specific recommendation either - but I did want to suggest seeing if any of the American study-abroad programs in Paris have suggestions. (Of course, this might work better during the academic year, though there are summer programs as well.) There are quite a number of US institutions that have programs there, plus the American University of Paris. Maybe emailing them will provide some help?

I say this because I did a fair amount of babysitting when I was a student in Paris. Unfortunately (for so many reasons!), that was long enough ago that I don't remember how I got the gigs!

There is an old thread on SlowTravel that discusses this:
http://slowtalk.com/groupee/forums/a/tpc/f/3956056284/m/1341089663 Perhaps some of the information is still accurate. At least it is a place to start.

We've had good luck at our hotel in both the UK and Spain. In one, the young woman who checked us in at the front desk mentioned that she had a flatmate from the US who was on a study abroad program and often babysat for guests. We met Lilly, a Boston College student, for lunch and she was perfect. She spent the entire weekend with us plus a few days when she didn't have classes and helped wrangle the kids. We still exchange Christmas cards with her.

The other good fortune we had was to have the daughter of the hotel owner in Spain, who also offered to babysit. We had stayed at this hotel several times in the past and had virtually seen this young woman grow up.

Times are different, though, so I'd probably be more careful than ever.