This is a phrase used to describe a mullet. The question is, did it exist before it was used to describe that hairstyle?
Sir Thomas Lipton from Google Books tells of the young Lipton sleeping in the back room of his shop. First in Britain, then America, shops had back rooms. These rooms could be used for a variety of things. I have known of many activities that went on in back rooms of shops.
The idea of "business up front, party in the back, must be very old. Pubs certainly might have had normal business up front with other sorts of "activities" in the back. This story: The Bowery Boys has an 1837 painting of events in the back room of a tavern.
A unique source for "business up front, party in the back" will probably never be found, but the idea must be very old. Maybe as old as buildings themselves.