And even if her husband had agreed to follow Kadohata aboard ENT, one could argue that having young children on a ship in times of war (and the Borg invasion(s) count as such) is questionable at best, even if the ships themselves may be designed for this... and you can't always evacuate them before an attack. I know I'd be exceedingly distracted if any children of mine were in our quarters (or whereever on the ship) and I couldn't be with them... I think even more so than if they weren't around on the ship at all.
Also, I didn't get the impression that the Ent-E was a family-ship like the D was, it's much smaller after all... Would it even have been possible for her family to stay on Enterprise, or would have Kadohata had to transfer to another ship in order to have her family with her?
Another question about Gold: Isn't he much older, therefore isn't his situation quite a bit different to Kadohata's? (Unless the SCE specifically mentioned that he either *now* has young children while serving as captain, or that he continued serving in the past even when his children were younger - and of course, it would be interesting, if there were any "family" ships around by the time his children were at the age of Kadohata's.)