Do you think the rising action begins with Rainsford's fall from the ship, or does it begin later in the story when he finds out about the hunt?

Ultimately, this question is up to you.  I have seen both answers adequately defended.  In my person opinion, I believe that Rainford's fall from the ship is the start of the rising action.  

The inciting incident of a story is the point in a plot when an event or decision begins that story's problem. Everything before this moment is introductory backstory.  I believe that Rainsford's problem begins the moment that he must go to Zaroff's island.  That occurs when he falls off of the ship.  The rising action continues to rise as Rainsford realizes that Zaroff is trapping and hunting people.  The action continues to build until the story's ultimate climax.