It is quite common for airlines to check for a credit card at check-in. This typically only happens on routes with high numbers of fraud or, in some cases, may happen across the whole airline. There is a way around this, such as booking with a travel agent (for instance).
Fortunately, I have never once been asked for ID or even asked to present a credit card used for the booking domestically with NZ. You may be asked to present ID in the instance that you are using someone else's fare and somebody has reported this (i.e. you "purchased" the fare from a Facebook page).