Quarter and core the cabbage, then thinly slice it.
Place the cabbage in a large mixing bowl, sprinkle with the salt, and toss thoroughly to evenly distribute the salt. Let the cabbage rest while you prepare the beet and ginger.
Shred the beet and finely dice the ginger. Add them to the cabbage and toss until evenly combined.
Transfer the mixture to a clean quart jar, packing it down firmly with a fermentation pounder as you go. Fill the jar until the packed vegetables reach about 1 inch below the rim.
Place a fermentation weight over the vegetables. Cover the jar with a clean towel and let it sit at room temperature for 24 hours.
Check that the vegetables are completely covered with brine. If needed, dissolve 1/2 teaspoon pickling salt in 1/2 cup water and add enough to the jar to keep the vegetables submerged.
Replace the fermentation weight and secure a fermentation lid. If using a regular canning lid, screw it on loosely enough to allow fermentation gases to escape, or briefly loosen it each day to release built-up pressure.
Ferment at room temperature, checking about every 2 days for surface scum and making sure the vegetables remain submerged. Begin tasting after 4 days and continue fermenting until the beet sauerkraut reaches your preferred flavor.
Remove the fermentation weight, cover the jar, and refrigerate the finished beet sauerkraut.
Notes
Ferment the sauerkraut away from direct sunlight in a spot with a fairly consistent temperature. About 65°F to 72°F is a good range for steady fermentation.
The beet will gradually color both the cabbage and brine as the sauerkraut ferments. This is completely normal.
Bubbling and a pleasantly sour aroma are normal signs of fermentation. Discard the batch if it develops fuzzy mold, a rotten odor, or other clear signs of spoilage.
Once refrigerated, keep the vegetables beneath the brine and use a clean utensil each time you remove sauerkraut from the jar.