New Zealand is generally a very safe destination, one of the safest in the world. Still, exercise the same care that you would in any major city. People-oriented dangers - theft, assault, murder - should be mentioned, but it's important to remember that violent crimes in most countries, especially in New Zealand, occur between acquaintances. As a traveler, it's unlikely you'll be a victim. If you're hitchhiking, however, that may be another matter.
On the subject of theft, it should be noted that many travelers are lulled into a false sense of security, leaving cars unlocked and valuables clearly visible. Always park your car in a well-populated area whenever possible; lock it and cover your luggage with a blanket or a coat. Never leave handbags or cameras in cars. The simple rule should be, if you can't do without it, don't leave it in the car, locked or otherwise.
The downtown areas of New Zealand's major cities, especially Auckland, Wellington, and Christchurch, are now well covered by closed-circuit cameras, which are monitored by police. This has significantly helped to reduce crime. It is still advisable to exercise caution, especially on Thursday, Friday, and Saturday nights, when social drinking sometimes gets out of hand.
There is a significant youth drinking problem in New Zealand, and this tends to manifest itself in areas with a large concentration of bars - Wellington's Courtenay Place and Christchurch's Oxford Strip are two good examples. That doesn't necessarily make these areas dangerous - far from it - but it is a good idea to be more vigilant and to stay away from any fights that might break out.
When it comes to traveling in the countryside, you should always be as cautious as you would be anywhere else. New Zealanders are generally friendly and welcoming, but if you're going to encounter any gender or racial bigotry, it is more likely to be in smaller rural communities. The risk is small, though. New Zealand is a bicultural country with increasing numbers of immigrants residing here.
Female travelers should never hitchhike alone or at night. Despite the wider population being tolerant of solo female travelers, there are always some unsavory characters who view this as an opportunity for theft, assault, rape, or worse. Don't put yourself in a vulnerable position. New Zealand men generally are fine, but there is a macho undercurrent in our society, especially when rugby and alcohol are part of the equation. Always take care in social environments, and if you are a single female traveling alone, don't drink too much, or you'll be easy prey.