I have been to Paris numerous times, one time right after the Gulf War. I have never had any problems. That said some basic things help.
Learn a few french phrases. Especially things like hello, thank you etc.
Don't expect it to be like home. People in general don't like being told that everything is better where you came from. (Yep, I have seen the Ugly American yelling about the coffee, pretty much a guarantee that they won't like you.)
Don't take the waiters personally. Yes, they can appear to be rude. The French don't get into the waiter introducing himself and making friends. Notice they are rude to fellow frenchmen also!
You'll be fine, the Parisians dislike Americans no more than any other race that isn't Parisian
Learn a few basic phrases in French, it's only polite, and getting the children to learn how to say hello, please and thank you always goes down well. Europeans are generally very keen on kids, mine get spoilt to death in places like Rome.
CarolA is right, what appears to be rudeness sometimes is just a different way of being - waiting staff being a prime example.
Paris now has several good Aparthotel types places which are great when travelling with kids.
I'm a travel journalist over here and have had various things published on Paris with kids, so if I can help...just ask
We went to Lyon and Paris in August 2002. The anti-American sentiment was much stronger last year than it was a couple of years ago. We actually had more than one person talk to us about politics. The French are even more honked off at the Americans and the Bush Administration now. Does that make France unsafe? I don't think so. Does it make the French even pricklier? Yup. The good news is that your kids will help break the ice somewhat. My 3 year old DD (she celebrated her birthday in Lyon) was our little embassador .