You don't need a license but you do need to check online whether horses are allowed on a particular beach first. Some beaches allow them but may have rules like you have to stay below the high tide mark or can only ride in certain areas.
I live in auckland and there are lots of beaches popular with riders like muriwai beach and karioitahi beach on the west coast, its good fun and you always bump into other people to ride with. These beaches also allow vehicles so it can get busy during summer with cars, motorbikes etc. Even ridden past a hovercraft which was interesting!