It's probably changed in the past few decades, but from what I understand it was kind of a "the club isn't the most expensive one you can join, but the money isn't enough to get you in" deal. Historically they wanted nothing to do with new money and especially disliked anyone in the entertainment industry. This is besides the standard you need to be a man, and protestant, and white, and when they say white they mean the Mayflower kind of white.
There is a story about Hugh Hefner building his bird sanctuary in retaliation for not being offered membership so there would be a ton of exotic birds constantly squawking off one of the holes.