Yes, they do. But fish have more luck in surviving the plunge than humans. They are better built to survive the plunge because they live in water all the time and are much more pliable and lighter than humans. At the bottom of the falls is a cushion of air bubbles that softens their fall. The body of a fish is also more suitable to absorb a tremendous amount of pressure hence, less injury in the plunge.
However, I often see birds circling the area below the falls hoping to grab any fish that might have been stunned by the journey over the falls.