Captain Barnaby is an old sea captain who takes players between East Ardougne, Brimhaven and Rimmington by his ship. His toll is 30 coins, similar to all other ships to and from Karamja.
This toll can be reduced to 15 coins by wearing any form of Karamja Gloves, obtained from the Karamja Tasks. You can also get a free ride back and forth from Brimhaven from Barnaby if you wear a Ring of charos(a).
During My Arm's Big Adventure he takes the player and My Arm to Karamja, After you have finished the quest, he'll start the conversation with: "Oh no, it's you again!"
After starting Dragon Slayer and learning about Crandor, the player has the option to say "I'd rather go to Crandor Isle." Captain Barnaby responds with "No, I need to stay alive, I have a wife and family to support." This dialogue only lasts until Ned agrees to help you navigate to Crandor.