I don't think you would have any problem with going to Dunn's River Falls from the pier or whatever. As always, I would SUGGEST that you do the tour through the cruise or a reputable touring company, but Dunn's River has so many people visit it that I doubt you would have a problem.
-Royal Caribbean Monarch of the Seas Southern Caribbean Cruise, August 2000
-Carnival Valor Western Caribbean Cruise, June 2005
-Sapphire Princess Voyage of the Glaciers Alaska Cruise, May 2008