Riders on subways, buses, and commuter trains should brace for steep fare increases, but extensive cuts in MBTA service now appear less likely as leaders of the state’s transportation system near a deadline to erase a projected deficit. Transportation Secretary Richard A. Davey, who has faced customers in a series of charged public hearings, said in an interview that his agency is in a desperate search to find revenue from other sources so the most severe service reductions can be averted.