Wordorigins.org

To launder money is to pass money from an illegal source through a legitimate business so that the illegal origin of the funds cannot be traced. The phrase can be traced back to the Watergate scandal of 1973, as evidenced from this citation from the Guardian newspaper of 19 April of that year:

Suitcases stuffed with 200,000 dollars of Republican campaign funds; money being “laundered” in Mexico.

But while the phrase can only be found in print from 1973, the metaphor of “washing” money is much older. There is this from the San Francisco Call-Bulletin of 3 June 1935:

There is not a hot money passer in America who will “wash” this money exchanging it for “cool” currency—unless it is offered him at such a tremendous discount that he can afford to hold it for years, if necessary, before attempting to pass it.

(Sources: Oxford English Dictionary, 2nd Edition; Safire’s New Political Dictionary