Probably an alt-Antarctic Treaty is set up to distribute who gets what, probably with some monetary transfers and other deals on the side. The big question is what the United States and the Soviet Union get, since this is the Cold War after all. Marie Byrd Land--over 1.6 million km2 of land--is up for grabs, so maybe the US and USSR could be convinced to split that?
There's also a
legal theory popular in parts of Latin America which states that nations have a right to the portions of the Antarctic directly to the south of their territory. The United States might like this, since this would give allied nations a sizable chunk of the continent. Unfortunately, most of that land is already claimed, but this revised treaty would ideally sort things out.
Overall, little activity is likely to take place outside of continuing scientific exploration. One big difference is there will be a lot more geological studies done on Antarctica and probably some low-level resource exploration along the coasts and offshore. We'll know a lot more about where the treasure troves of Antarctic resources are. However, barring huge problems in Latin America and South Africa (key exporters of certain resources which could have major civil wars), I don't think anyone would seriously consider mining them due to their otherwise lack of profitability.
Antarctica might have a few more places for tourists, but no one is likely to live there permanently. Building a prison/gulag in Antarctica also won't happen, since it's too expensive and nations like the USSR, Latin American dictatorships, etc. already have plenty of remote places to put prisoners.