Yes you can. you can grab a cab to take you to Red Hook ferry and it will take you across to St. Johns.
Life is too short to let the ship of your dreams sail without you.
Carnival Destiny Feb. 2006
Carnival Fascination Feb. 2007
Royal Caribbean Grandeur of the Seas Feb. 2008 The DTW & MsBJ tour
Carnival Valor MsBJ and Dina Feb. 2009
Carnival Glory MsBJ and Dina Feb. 2010
Carnival Dream MsBJ and Dina Feb. 2011
There are ferries, but booking an excursion does give you a chance to tour the island, unless you just want to lay in the sun or go on your own. Do be warned that you will need to keep check on the clock by going it alone.