Cheapflights.co.nz has researched which airports in Australia and New Zealand offer free WiFi:
Auckland: 30 minutes of free WiFi is available from hotspots in international and domestic terminals.
Christchurch: 30 minutes of free WiFi is available from hotspots in most public areas of the terminal.
Wellington: Free WiFi is available from the food court area on level 1 of the main terminal.
Queenstown: Free WiFi for one hour per every four-hour period.
Dunedin: Free WiFi for 30 mins or 1000mb (whichever occurs first).
Adelaide: Free WiFi at hotspots throughout airport.
Brisbane: An hour of free WiFi is offered in both the international and domestic terminals. The service is available in the retail areas of both terminals and there is a download limit of 50mb per user.
Cairns: $3 per hour/60MB, $6 per 24hr/160MB or $30 per 7 days/1.2GB. Internet kiosks are available at the same cost as WiFi.
Darwin: Free WiFi is available on the first floor of the terminal.
Gold Coast: Free WiFi is available throughout the terminal
Melbourne: Free WiFi in all public areas and aircraft gate lounges in all terminals. Gates 12-20 in Terminal 2 have reduced signal strength.
Perth: There are several free internet booths in the international and domestic terminals. There is free WiFi in Terminal 2 (international) and Terminal 3 (domestic).
Sydney: Free in both the international and domestic terminals. The service is available free for up to two hours and allows 500mb of data per user.
Ex NZ Herald 24 June 2014