Yes I believe it does run. However, be aware that there are sign up deadlines for port days, so if you do not sign up your kids for the program prior to the deadline and no one else does, the staff will take the time in port off. Also be aware that NCL charges hourly for the kids club on port days which is different than many lines however that didn't bother me too much since you pay reduced tips for the kids on NCL, so it all kind of evens out. The 2 year old will love the kids club, be aware though that you will be given a beeper in the event the toddler needs a diaper change (if not already potty trained) you are required to come immediately and care for the diaper change.
Now this is one area where Carnival is a bit more accomodating than other lines (aside from Disney, but they are pricier) Carnival will babysit the one year old until noon on port mornings if I recall correctly and they also change diapers. They will give you a wireless phone or a pager and call you if needed, but only if its an issue they cannot address.