Far fewer than most people assume, though the number has grown sharply over the last decade or so. Antarctic tourism is small compared with any mainstream destination, and the growth has come from more ships being built and more people treating the seventh continent as reachable rather than theoretical.
Scale is worth holding onto when you read about crowding. The 100-ashore rule means that however many people are on the continent in a season, no more than 100 stand at any one site at any one time, and sites are booked between ships so two vessels aren't at the same beach.
We're deliberately not quoting a precise annual figure here. The numbers get republished from stale sources constantly and we'd rather send you to IAATO's own current statistics than repeat something we can't stand behind. Ask us and we'll point you at the live source.