You might, and it depends on your passport rather than on Antarctica. Most Peninsula voyages leave from Argentina or Chile, and some East Antarctica voyages use Australia or New Zealand. Argentina and Chile both apply nationality-specific visitor rules. Most non-Australian passport holders need the correct Australian visa before travel. Many visitors to New Zealand need either a visitor visa or an NZeTA, plus the New Zealand Traveller Declaration on arrival.
This is worth raising early rather than late, because it can change which trip suits you. If a Chilean visa is difficult for your nationality, that pushes you toward sailing from Ushuaia. If Argentina is the harder one, flying from Punta Arenas solves it.
One thing that catches people: you can need a transit visa even without leaving the airport. It depends on the country, your passport, the terminal, your baggage arrangement and the length of your connection. Australia warns about this explicitly. Check every transit point in both directions, because an airline can refuse boarding without it.
Pre or post cruise travel through Peru, Bolivia or Brazil can also add requirements, including yellow fever certificate considerations for a later destination. Build one document checklist for the whole journey rather than just the cruise.
Ask us which countries your itinerary touches and we'll tell you who to contact. The determination itself has to come from them.