The only ones I am aware of in Corsica are in Calvi, at the NW corner of the island.

I've frequently seen unattended boats spending time there, but cannot comment on prices for long stays.

Prices for overnight stays are about 2/3rds of the nearby marina (which is chock-full in season).

There are ferrys to Italy and France, and there's an airport nearby as well.

It's quite far from the Balearics though (about 300-odd miles from Menorca), and it may be a large detour if you plan to head south afterwards.