My wife and I used Sorrento as our home base (a very beautiful and charming town). We would take day trips to all the islands, Capri, Positano, Amalfi, Ischia, etc. So we only had the one hotel, which was right in the center of town. (I'll have to ask the wife for the name of it). But having been to Italy many times, I would say every hotel I've ever stayed at has been kid friendly. There are so many people on motorbikes day and night, lets just say you'll be hard pressed to find a very quiet spot (though I'm sure they exist there, if you're not in the city). I'd imagine the islands would be much quieter due to many of the streets not having motor access, or very little compared to Sorrento or Rome, etc. Rick Steves has a nice tourism video on the Amalfi coast. You should rent it from the library for ideas.
Have a great trip. I will return one day when my two girls get a bit older myself. It truly is heaven on earth over there.