Antony Beck, president of his family’s Gainesway Farm north of Lexington, Ky., sees himself as a steward of some of the Bluegrass country’s most historic and productive land. But the 125-year-old farm isn’t a fusty old museum relic. Inside its unpretentious, old-fashioned office and its graceful barns, Gainesway hums with the purposeful activity of a modern Thoroughbred business. That’s something Beck, 48, takes seriously: keeping the operation commercially successful, especially during the bloodstock business’s recent downturn.