More often than not, I would bet that the predominant policy is that if their money is green, plentiful and timely, and that they behave "appropriately" and below the public glare while at the club- they are fine.
I know of at least one such member at my home club presently, though there are probably several others with very interesting backgrounds. A former member who used to get plucked regularly to the tune of thousands at a time left the club unexpectedly. Several years later someone posted an article in the men's grill about this guy who apparently had been sent to federal prison for a number of years for among other things, selling high-risk commercial insurance and not turning in the premiums. That appears to have been the source of his funds for gambling (a terrible golfer he was, though he played nearly every day). The irony is that while in prison, our member found his true calling and published two highly successful crime novels. I don't remember his name, but I know that the gamblers are counting the days until he comes back. I have no doubt that with a small initiation fee and $315 a month, our felon would be welcomed with open arms.