Yes, meaningfully. Stabilisers take a lot of the roll out of a crossing and they're one of the reasons the Drake has a gentler reputation among recent travellers than it did a couple of decades ago. Hull design matters too, and the two together change how a given sea state feels in your cabin.
It's a real factor but not one worth over-weighting against everything else. Every expedition ship crossing the Drake is ice-strengthened and stabilised to some degree. The differences between them are smaller than the difference between a calm day and a rough one.