For traditional and authentic borscht, the standard, time-tested choice is classic white head cabbage (Brassica oleracea var. capitata), commonly sold in North American grocery stores simply as green cabbage.
In Eastern European culinary tradition, particularly Ukrainian cooking, this is known as white cabbage (bilokachanna kapusta). It is prized for its dense leaf structure and balanced sweetness when simmered. When selecting a head, especially for a large batch intended for freezing, you should look for a firm, heavy head typical of late-season harvests. Late-harvest white cabbage has thicker cell walls and lower water content than delicate summer varieties, which allows the shredded ribbons to retain some bite and structure even after being frozen, thawed, and reheated.
There is one other historically authentic variation worth noting: naturally fermented cabbage, or sauerkraut (kvashena kapusta). In traditional rural households during late winter and early spring, fresh cabbage would often run out or lose quality, so cooks would make winter borscht using home-fermented sour cabbage. This served a dual purpose by providing the vegetable bulk while also introducing the natural acidity that every authentic borscht requires.
Modern recommendations like Napa cabbage or Savoy cabbage are indeed ill-suited for this purpose. Napa cabbage contains far too much water and delicate leaf surface, causing it to disintegrate into an unappealing pulp after freezing. Savoy cabbage has a crinkled texture and earthy aroma that alters the profile of the broth and softens too quickly.
To get the best result with standard green or white cabbage, shred it by hand into uniform ribbons about a quarter-inch wide. Add it toward the latter half of cooking so it cooks through without turning overly soft before you portion and freeze your containers.