Flip Flop Fly Ball heeded the call from the Voice in Field of Dreams and discovered which Red Sox game Kevin Costner's and James Earl Jones' characters attended in the film.
The blog tracked down which game it was by comparing the lineup in Costner's game program with the actual Red Sox lineups used against their opponent, the Oakland A's, in 1988. (The film came out in 1989, so it stands to reason that they filmed it the year before.)
The date? May 16, 1988.
Unfortunately, the props department on the film must not have been too studious. Flip Flop Fly Ball notes that the scorecard Costner filled out does not match up to the actual game outcome.