There is train service between Fort Lauderdale and Miami (and West Palm Beach, too). However, this is mainly for commuters. It does not run in early afternoons, for example. You can go from the Fort Lauderdale airport station (via a free shuttle) to the subway connection station, to the Government Center station, and then catch a shuttle to the port. It is rather convoluted and takes a little over an hour, but it is possable.