Yes, they do. Most survive the hard journey over the falls, sometimes laying stunned for a few minutes at the bottom before continuing their journey down the river.
Since water is the natural habitat for fish and they are much more buoyant than we are, going over the falls isn't much different for them as an elevator is for us. Most of the time the fish stays in the water as it heads over the falls, and the foam at the bottom acts as a cushion for them. They can withstand a large amount of pressure before it will cause any injury to them. The pool at the bottom of the Falls is about 150 feet deep so most of the time they aren't forced to the river bed. Their biggest threat when going over the Falls isn't the fall itself but the seagulls waiting at the bottom to pluck the stunned fish out of the water before they get the chance to recover.
More fish are injured going over American and Bridal Veil falls due being crushed against the rocks at the bottom of the Falls.
Interestinly enough, a number of years back a tourist was hit by a salmon while taking a stroll on the Cave of the Winds boardwalk. He wasn't hurt, but he did catch the fish and was allowed to keep it.
You won't see fish jumping out of the water at the bottom due to the volume of water falling, and it's very rare to see dead fish at the bottom.
What kind of fish are found?
The Niagara River contains around 94 species of fish including the Chinook and Coho Salmon, Smallmouth Bass, Rainbow and Lake Trout, Catfish Muskellunge, Northern Pike and Yellow Perch. The biggest fish found in the Niagara is the Lake Sturgeon that can reach 7 feet in length and weigh more than 300 pounds. It's not unheard of to catch a salmon weighing 20 to 30 pounds during the fall months