I am in Nelson, so cant' speak for the other cities. Here dogs don't seem to be a problem at all. We often see dogs on and off the leash, being walked along the rivers, the islands, etc etc. We have 2 puppies and one older dog.
There are a lot of media articles about kids being mauled by dogs in the face. Mostly these are pitbulls, I think.
We have a lot of pig hunters here in the Nelson / Marlborough region, so there are always utes riding around with pig dogs in cages on the back. My old dog is an ex pig hunter. As gentle as anything.
I personally think that because it is a small country, and EVERYTHING is reported, it seems to be a problem , but is not.