Actually, you may find some activities scheduled on NCL during port days. Not as numerous as on sea days, but there are usually some quiet things going on... movies, exercise classes, tea, and so on. Of course the spa is open and there are usually discounts on the treatments on port days.
NCL doesn't necessarily close the children's centers on port days. If you want your children enrolled on a port day, you must sign up by 8pm--it's considered "group babysitting" and there is a charge.
Hope this helps.