“…. ‘Redman,’ who, with all his eccentricities, is the best second base player in Hoboken, and one of the most fearless base runners we know of.” The New York Times, October 1, 1865, p8
Previous earliest use (Dickson Baseball Dictionary, 3rd edition, 2009):
1868. “Base Runner.—The player on the batting side running the bases” (Henry Chadwick, The Game of Base Ball, p. 38).