My kids are playing Animal Crossing New Horizons and Kapp'n won't take us on his boat. I have verified and they have enough Nook Miles (over 15k).
We are not playing in English but if I translate freely he says something like: "I only take old sea wolf. You're too inexperienced. If you still want a ride go the airport first.".
Going the the airport we have the options for multi-player, going to Joe's island and something about buying a Nook ticket. We tried going to Joe's island. We waited a day in real life and tried again.
I'm not super familiar with the game. Searching online and it seems you just have to go up to him and pay the fee.