Take your typical money laundering scheme. Criminal J has some illegally-acquired cash, entirely off the books. He wants to launder it. What he really want is to give it to someone, who will then give it back to him, under the guise of some legitimate-seeming transaction.
So, Criminal J deposits the money in Assistant's account. Assistant then "buys" something from Criminal J. He never actually delivers those goods, but this transaction is now on the books, looks entirely legal, and Criminal can deposit that money into his own account along with normal business.
But, someone looks at Assistant's books. And there's a bunch of money, the origin of which is dubious - Assistant has more cash flow out than they take in. Assistant then clearly has some off-the-books income, which the Cops want to figure out. If that deposit was anonymous cash, the trail stops at Assistant. If the bank requires that cash transactions come from known accounts, then the trail goes back to Criminal J. Does Crtiminal J want that? Heck no!
So, again I ask - if you are doing something financially hinkey, do you want your real name, address, and SSN attached to it? Probably not.