I don't have any kids, but a good friend of mine has the same irrational fear of her kids falling over the side so when we were on our first
disney cruise, we did a walkabout on the ship to see what we could tell her to make her feel better.
First off, there is definately a lot less "open railing" than I ever expected on the cruise ship. You actually need to go looking for it in the public areas, and of course there are a ton of verandah rooms on the ship.
Deck 4 is one of those places, but there is nothing out there for kids to do. It's basically a place to sit and walk and play shuffleboard. We only went there once... for the safety drill where we were all wearing lifejackets and packed in like sardines and there were so many staff around I don't think a kid could get near the railing, let alone climb up it. There's really nothing for kids to do there.
Deck 9 is the deck where all of the pools are. While they are completely open to the sky, all around the edge of the boat right above you is deck 10... kind of a balcony overlooking the pools. In between deck 9 and deck 10 is full plexiglass, kind of like a greenhouse. THe only place on deck 9 where there are open rails are all the way forward (where your kids should not be anyway because all that is there is the adult pool and the spa) and all the way aft in the outdoor seating area of topsiders, the buffet restaurant. There is also plenty of comfortable indoor seating in topsiders, so again you could easily avoid that area all together.
The only thing on deck 10 is another walkaround area that overlooks deck 9, Palo's restaurant and the wide world of sports, both of which can be accessed without going outside onto a deck where there are railings (though Palos is not appropriate for kids, don't know about the sports thing).
You can also get a cabin with a porthole rather than a verandah (or even inside) the porthole is not operable so the room is totally sealed, nowhere for kids to fall out into.
So basically, unless you want to be near a railing to look over, there is no reason to have to go there (no activities or meals that force you to). I would not have noticed if I hadn't been looking for it but basically Disney definately had the little ones in mind when they designed the ship. None of the kids activities (where they might be running around and excited or otherwise) take place anywhere near a place where there are only railings. If you DO go near railings, they have plexi on the inside to prevent kids from using them as ladders. This ship is really more like a land resort than you might imagine, except more arranged vertically than spread out and horizontal.
As far as being afraid of heights, the expanse of the ocean really messes with your perception of how high up you are. Unless you are hanging over the railing, you really can't tell how high up you are because all you see is the horizon pretty much where you expect it to be. I have always been a little nervous around heights and after a bad experience 1000m up on the cliffs of Capri I am downright terrified around heights but up on Deck 10 with my chin on the railing was one of my favorite places to be!
I hope that helps alleviate your fears somewhat!
Lisa