I was waiting to see if someone with experience would answer, but they haven't. The seas in the Atlantic Ocean from NYC to Bermuda can run from smooth to very choppy -- it all depends on hurricane activity and other storms on the water. You best bet would be June - September for the smoothest seas (that is barring a hurricane).
Princess does have a childrens program, and I would think the same would hold true on all their trips.
Now, I took the Pacific Princess to Bermuda in 2000. She is a great, small ship with typical Princess detail. Went in July and had no problem with the seas.
Now, for Bermuda, choice between the Nordic Empress and her when having children on board would be hard. Nordic (just returned from her last night) has a nice childrens room, etc. Princess does offer the kids program dependent on the number of kids on board I believe.