Most of the balcony railings are shielded with plexiglass (not so on the aft staterooms, which are solid), but even so the view through the glass is a bit unclear, and our DD5 often wanted a better view of things. I noticed her looking at the small plastic table on the balcony, and I had to advise her very sternly that under no circumstances was she permitted to try to stand on that table or either of the chairs to get a better view over the railing.
As Lloyd suggested, if you're supervising your children I cannot imagine any problems -- I definitely wouldn't leave "little climbers" out there alone for any length of time.
That advice goes for the rest of the ship as well -- I could be wrong, but don't think the promenade along Deck 4 has the plexiglass, and those railings could easily be climbed in just a few seconds by a child. The railings along deck 10 up are much higher, so they wouldn't be a problem.