One train per week stops at the station (every saturday morning I think) in each direction, I think the reason being that it is so far from the terminal. They could set up a minibus/bus service to the station from the terminal.
The other way of getting to the airport is by bus, with a bus every hour from Darlington, and a bus every hour from Middlesbrough. These buses could be extended to serve the rail station.
HTH
mmeteesside
MME Movements