We always talk about down south and up north. One usually associates going south with going down. In the sentence that you have given, south means down. The demand for the dollar has pushed the value of the rupee further down.
In American English, the expression go south is used in informal contexts. When you say that an organization has gone south, it means that though the organization was in excellent shape once, it is in extremely bad shape now.
COURTESY : The Hindu (The National News-Paper) - India