Continuity mistake: In the same scene where Brooks is hanging himself, he carves his name and he is almost eye level with the spot where he is carving his name. The place where the rope is tied is lower. How come when he finishes carving his name he stretches to put the rope on? It is lower than where he was carving his name. (01:00:25)
Continuity mistake: In the two different shots we have of Tim Robbins getting ready to put the book and folder into the safe for the last time (before he escapes) he collects the book and folder in two very different ways. First time he pulls the book towards the folder. Second time he brings the folder to the book. (01:49:55)
Continuity mistake: When Hank Williams is singing "Lovesick Blues" the MGM label is from 1968. This is meant to be 1963.
Continuity mistake: The warden pulls out a box of Remington .38 Special cartridges to load his revolver prior to his suicide. The warden empties the cartridges on to his desk and loads the revolver with what are Winchester/Western cartridges.
Chosen answer: He probably only knows the name. He doesn't want to know anymore, so he can pin the blame on Andy should anything go wrong. He no doubt hasn't even considered that Andy might have ID so he can access the accounts after his escape.
Grumpy Scot
Also, Andy indicated that the deposits were night deposits, which were in envelopes. So, the warden may not have known the "Randall Stevens" name.