It's a question audiences have been asking since 1933, so I thought this would be the perfect place for it :) Even Peter Jackson decided to side-step this particular issue in his remake.
How did they get "King Kong" back to New York?
The 1933 movie doesn't show how King Kong was transported, but according to the script:
They possibly towed the raft with the ship, all the way to New York.
(Maybe Peter Jackson thought using a raft is a silly idea and therefore didn't include it in his version)
However, the raft idea was used in King Kong vs. Godzilla:
In the 1976 version they used an oil tanker to transport Kong: