I went aboard ship with my parents when me and Sis were 4 and 2. It was doable. No kids club in those days, Sis and I were the only children so had very high standards drilled into us.

Seasickness... those seasickness bands are fantastic. Get some. We never got seasickness - when you're younger it's easier to adapt to the movement of the ship so long as there aren't any of those ghastly stabilisers on the thing - so can't comment there. If your children drive you batty too often or don't listen to safety things (ie won't hold hands while crossing the road type of thing) then I'd probably worry about falling overboard too. I know my mum was very paranoid about it. But not too stressed, she did enjoy herself too.

DH doesn't like ships whatsoever so my cruising days are over for now, but I'd go with DS even though he's a handful. There's so much to learn on a ship! I learnt all my astronomy on a ship off the coast of Aus and NZ - so pretty useless in the UK as I don't know our stars as well as I do yours LOL.