Have you tried the borscht at the Russian Tea House? I can't eat beets, so I can't report on it, but their other dishes are very nice. The Russian Tea House is a very casual, mostly take-out place in the Midway area of St. Paul.
Another place to check out: Moscow On the Hill. They definitely have borscht on the menu. Me, I love the salmon blini (which is more like a crepe, but is very tasty).
Kramarczuk's has borscht, too. This is a Ukrainian/Polish deli, so it might not be the same as Russian-style borscht. When I'm here, I get the Szegedy Goulash, which has a Hungarian name but is clearly Slavic in taste.
Kramarczuk East European Deli
215 E Hennepin Ave, Minneapolis, MN 55414
Russian Tea House
1758 University Ave W, Saint Paul, MN 55104
Moscow On the Hill - Duplicate
371 Selby Ave, St Paul, MN
I haven't had it, but here's an excerpt from a review of Nick and Eddies, a popular place in Loring Park.
For four bucks, there's a generous bowl of borscht, which starts with a base of beef-cheek braising liquid, beets, and cabbage, and finishes with a whirl of dill, sour cream, and a kick of smoked paprika and chili.