GCS: I'll let others answer the children part except to say a destination like the BVI would be good given short sails & many options for beaches + other activities.
The food for thought I would offer is regards to sailing. There will be a specific amount of attention & crew help that will be necessary to sail safely
no matter where you go. You also have to factor in the Dinghy which will be used everywhere you stop. So keeping a eye on a small child when on the boat (moving or not) and transferring in/out of the Dinghy has to be factored into your decision.
I have not heard of any "child friendly" charters or charter companies but you do see children on boats.
One alternative idea where children are welcomed = Sunsail Clubs. Mom & Dad can sail (smaller boats) and or enjoy their vacation with help avail for your child.
I sailed & drove by the club during a recent charter + had a buddy who did a vacation there and said there were lots of families/kids. It's another place to scan lower rate seasons to get best deal.