Here is an excerpt from Penn State's Vegetable Gardening book:
Cercospora Leaf Spot of Beets
Symptoms are spots that are brown or gray with a purple border. They may develop late in the season. Most table beets are resistant to this disease. Severely diseased plants can become defoliated.
Management: Follow a 2- to 3-year rotation (i.e., don’t plant beets in the same area for 2 or 3 years). Remove refuse from the garden as soon as beets are harvested. If necessary, a pesticide that contains fixed copper may help; follow the directions for mixing and application.
And here is a link to that informative book: