We spent two weeks in Corsica in September 2024. Loved it. But I think you are going to find some things about your plan to be, um, challenging. Right off the bat:
We are planning on a four night visit to Corsica in January.
January. Well, at least you won't have to worry about crowds!
We won't be going to the beach for swimming
I believe you! And I hope you have an accurate idea of what to expect for weather in January. You won't be the only ones staying out of the water. It's not going to be warm enough for anyone to swim. In mid-September, we found the water temperature tolerable, but not really comfortable. We went swimming once, briefly, and that was enough.
As for a good "base" it's a bit tricky, because the places we found most interesting were inconveniently scattered all around the island.
But you only have 4 nights. Four nights gets you just three full days. With just three days, you don't really have time to move around much (or at all). What's the thing about Corsica that most interests you? If it's wine from Cap Corse, then stay in either Bastia, or maybe Saint Florent (like Bastia, it's sort of at the base of the Cap Corse peninsula, but on the western shore). Have you checked to see if any of the Cap Corse wineries are even open in January? You might want to look into that.
A lot of the island's tourist services will be shut down completely in January (for good reason). As a holiday town, we found Saint Florent charming, but that was in early September. I suspect it would feel pretty dead in January (but that's going to be true of most of Corsica).
About those airports: Bastia gets the most flights, but all the flight schedules are highly "seasonal" (meaning, flights get very limited outside of high season, many flights only run in summer - check your assumptions for what flights will actually be going in January).
Our favorite place in Corsica was Bonafacio - but that's at the opposite end of where you are looking.
Agree with the post above that it takes longer than one probably expects to get around - roads are slow-going and twisty. We loved our 2 weeks in Corsica, but I would have to think twice (and then think again) before going there in January. I also feel that Corsica is worth a lot more time than just 3 days (we had 2 weeks, but wished we had had 3 weeks).
Hope some of the above is helpful. Good luck.