"I have found ASB, especially since they upgraded their Fastnet webbanking a couple of weeks ago, to be really good in this area. I've never had browser problems."
"Ditto the National Bank."
"Kiwibank also has great online banking, and have been very receptive to the emails I've sent through pointing out faults in their website (mostly incorrect/incomplete documentation)."
2 pages link to LinuxFriendlyBanks: