You can look at a map of the districts, if I had my choice, staying with kids, I would probably stay near the Sorbonne/Luxembourg Gardens, in the 5th. It's a fun area, with lots of quick places to eat (good for kids) and accessible to everything without being overly touristy. Good metro access, good walking access. I have stayed in the 2nd, near Les Halles, and I wasn't crazy about that area, a little dodgy at night. I have stayed in the 7th, a nice quiet upscale area, but a little far from the sights. 15th and 18th are nice and fairly convenient, but you probably want to be someplace closer to the center if you have kids with you.
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Also btw, I was in Paris in September, and the apartment we rented had bedbugs! This is a huge problem in New York, so I had been reading scary articles for months and identified the problem immediately, but it's only starting to be a problem elsewhere, so there's not as much awareness. Anyway, you might want to read up on what they look like and how to check for them before you go, because it's not something you want to deal with or bring back home.Â