For Alaska cruises (Salmon fishing), if you want to keep your catch, you can pay to have it "prepared" and shipped to you. You can not bring them back to the ship. Not sure how the arrangements were made for the post above about the chief preparing a catch.
The cost of doing this was more that just stopping in the port and having commerically caught Salmon shipped to you.
For the Caribbean, its the same thing, you can have a big fish shipped to you (can't bring it back on board with you) but the costs are high.
LHP has it correct. The crew will more than be willing to take it from you as they can turn around and sell your catch on the dock and make some extra money. But catch and release is a good alternative too.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007