Sorry Maddie - I did wonder if the link would only work if you were logged into Ancestry. Guess that's one way they persuade people to join!
Steve, brick wall, I'm afraid. We checked the national bmd, and as already reported by someone else, nothing. The archivist there told me the national records from 1888 are pretty good, but we checked the local daily newspaper for Dunedin - nothing. Also checked all post office & trade directories for that decade, nothing (even in the all-NZ directory).
Checked museum archives for him or his parents - nothing. There is no comprehensive index for passenger lists though so no way to know when his parents arrived. Incidentally, if you ever look into this again, there was a Walter Bedford in Petone/Wgtn area (in school records in 1880s/90s, so not your Walter) but his dad was George, so a red herring.
Couple of possibilities:
The archivist told me births on board ship at this date would have been registered in the country in which the ship was registered. So if they came out from England, his birth would be registered there even if he was born as the ship was about to dock in Dunedin.
Also noted that if that passenger listing is him (we didn't find any other S Bedfords in our searches, but that's not to say there wouldn't be any) then he left NZ from Wellington, not Dunedin. So someone in Wellington might be able to check directories etc there for you.