I'm not sure if the general public can lauch there, but at Stony point, Betty's Bay, there is a boat slip. It does have a gate / boom, so I'm not sure if you need to be a resident or member of something to use it or not ?
It's right next to the penguin colony boardwalk, but I'm not sure how you would find out if you may use it or not, it looks fairly new - it's built right next to the old slip. Parks board might know, as they charge a fee to view the penguins, so they have a office right next to the slipway.