Blindfolds just seem unnecessarily humiliating, and not all that secure. Blindfolds can come loose. If it's really that important something stay secret, put it behind a wall.
Humiliation would be if they mocked you for being blindfolded or once you were blindfold stripped you naked and paraded you through the canteen. It's just part of the process, everyone in your position goes through it and no one there would see any humiliation in that.
No matter where you put that wall, there's a chance something on the other side of it will need servicing.
I'm sure the facility that makes nuclear warheads has an electronics inspection x-ray. If they need maintenance on that, you'd expect him/her to have a detailed background check, and to be shown as little of the facility as possible, and not to be allowed to leave with USB memory sticks or suchlike.
Of course, who knows how many facilities really need that level of security, and how often it's paranoia or theatre?
