Well, first of all, most of the staff are pretty insane themselves.
First off, lest we forget, Harley used to be a psychiatrist working there.
Then there was Dr. Hugo Strange, and Dr. Jonathan Crane (aka the Scarecrow), both former doctors at Arkham.
Lest we forget Dr. Jeremiah Arhkam himself. At first he was an okay and well-meaning shrink (but had the worst job one could have, running Arkham) only to go off the deep end himself and became the new Black Mask. (Fortunately, he seems okay in The New 52.)