There is a water taxi between north and south Bimini. The regular airplane airport is on South Bimini. The cost is apparently $3 to $5. I have not taken it, nor was I ever sucessful in seeing it although I heard the water taxi being called on the VHF many times. The cost may be included in packages for Bimini Blue Water resort if you fly into BIM (which I think is the regular - non seaplane - airport on South Bimini)
The mailboat runs weekly between the islands and the fare is $35 one way. The Bimini/Cat Cay mailboat is called BIMINI MACK. .