Violent crime is not common in NZ and although Auckland is considered the 'crime capital' of the country, it's very safe by most international city standards. Theft, primarily from cars, is a problem so avoid leaving any valuables in a vehicle no matter where it's parked. The worst places to tempt fate are tourist parking areas and the car parks at the start of walks.
Don't underestimate the risks posed by NZ's unpredictable, ever-changing maritime climate in high-altitude areas.
NZ doesn't have the proliferation of venomous creatures found in neighbouring Australia (poisonous spiders, snakes, jellyfish, etc). Sharks exist in NZ waters but they are well-fed by the abundant marine life and rarely pose a threat to humans; that said, attacks on humans do occasionally occur. Much greater hazards in the ocean are the rips or undertows that are capable of dragging swimmers right out to sea - take notice of any local warnings when swimming, surfing or diving.
The islands' byways are often made hazardous by speeding locals, wide-cornering campervans and traffic-ignorant sheep. Instead of careering around the country at top speed, set yourself a reasonable itinerary and try to stay alert on the road despite the distractingly beautiful scenery.
For sheer annoyance, it's hard to top a sandfly visitation. Equip yourself with hardcore insect repellent in coastal areas unless you're keen to imitate a whirling dervish when these little devils start biting.