Gee, we do that all summer, not far from NY too, with 4 kids ages 5 to 13, one with autism. And we could house a cameraperson in the crew quarters.
Problem is, that every minute of the summer is priceless. Not sure we'd want to take the time to deal with all the cameras and everything. And the secret spots we go to are special -- I'd share them on this forum without hesitation, but sharing them on prime time seems different somehow.
Each day is different and dynamic and serendipitous -- on the inside. For an external viewer, not sure how much a viewer could stand watching us 1) playing in/on/under the water, 2) playing/chatting on the beach, 3) reading, 4) deciding what to do for dinner, 5) choosing the venues for each part of the day, and 6) teaching the kids. And sailing, of course. What's missing is back stabbing, scheming and other things that seem to make ratings. Instead it's beach, bar-b-ques, bon fires, wine/beers with family and friends, sunsets, and simple fun. It's basically a moving, relaxed-but-careful party in a big playground. Like I said, it's summer.
Also, any livaboard is going to be smart enough to ask to see your previous work, to see if you are creating ratings or actually showing what life's like. Think of a scale between Ken Burns and Jerry Springer. You might want to post links to some of your previous work.
Good luck in your search. If it has some good sailing clips, I'm sure we'd all want to watch it.
Great minds discuss ideas;
Average minds discuss events;
Small minds discuss people.
The best minds discuss sailing. I don't know why. It's a mystery!