I think you would be better off doing a taxi. IF you do stay at the airport, many/most of the hotels offer shuttle service to/form the airport. Then you would only have to pay for a taxi to get to the port (figure $35 with tip for a van sized taxi) So you get 2 or 3 taxis from the hotel to the port. (This is our plan for Miami in a couple of weeks)
BUT even if you decided to stay downtown, the extra taxi ride from downtown to the ship would only be $15 to $20. The cruise line charges per person not matter how many people you have. The taxis charge per taxi load. With ten people the per person cost can quickly out strip the per taxi cost.
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, Celebrity Silhouette 2013