In the song "Knoxville Girl" the singer exorts the girl of the title to "Go down, go down you Knoxville girl...." after throwing her into the river to drown. In that case, the meaning is quite obvious and literal: "Sink below the water's surface."

I would make an uneducated and highly speculative guess that the term may have made its way into secular song via the "Go down, Moses" exortation heard in spirituals. Its meaning in that case is "Go down (south) from Israel into Egypt."

As Leenia stated above, it's never been part of day-to-day speech, but is strictly a songwriting convention. Its exact meaning varies from song to song, depending on the context.