have done 4 cruises with a little girl aged 2, 3, 4, and 5
On P&O (2 auroram 2 Oceana)
you cant fall off the side.... even if you were trying to jump into the sea i think you would find it extreamy difficult.
Even when it is rough it isnt actually that rough aboard if you see what i mean?
the ships have stablilers and most of the time you dont know you are moving at all... even the drinks in glasses dont ripple. (i have never done transatlantic though, but have done norway, baltic and carribean)
ONE time for about an hour the decks were closed because apparently it was too rough and dangerouse but even then it was still ver flat inside.... you just occasionaly might hold the hand rail to steady yourself.
That is ONE day in over 9 weeks of being on a cruise ship!
Cruises are great IMO
(as for the heacvy doors to the deck, on board Oceana (and i cant remember about Aurora) lots of the doors were automatic so a child could go through.... but you really could NOT fall over board