I am moderately claustrophobic (have to close my eyes and breathe deeply in crowded elevators, prefer sitting in an aisle seat where there's more "breathing room" on airplanes, etc.). DH rode on Mission: Space before I did and encouraged me to go ahead and ride, and I'm so glad I did! I was very nervous, and I did have to close my eyes for a few seconds when they first closed the door and the control panel closed in on me, but after that I was fine and was able to thoroughly enjoy the ride.
That being said, I'm sure your response would really depend on your level of claustrophobia. I imagine that someone with severe claustrophobia would definitely not be too happy on this ride!