I'm posting here because I'm having the same issue and want this topic up top. I finished the story. I found the dock from which you would most likely embark. You can't land a helicopter on it. You can't swim there. Boats are repelled. It has become the bane of my Lego City experience.
You will need to have completed Story Mode, and have Rex Fury equipped as your Criminal disguise.
In the Downtown district, seek out the round fortress. (It has planters spaced around the walls outside -- on the side nearest the waterfront, there's an entrance in the wall.) Use Rex Fury to pull apart the big crate with the orange handle in the center of the area, then put together the pieces you find inside. You'll have a green pipe that ought to be very familiar to practically everyone. Stand on top of the pipe and press A. Voila!
After you unlock Liberty Island, you can build a ferry at the docks, which will take you to a ferry point in Crescent Sands (? It's Crescent something, but I don't have the game open right now), and allow you to travel to the island that way afterwards.