The passenger terminal was close to the station in 1971. Then the airport owners relocated the passenger terminal to a different part of the airfield that is quite some way away from the station, which rather shows how small a proportion of air passengers started/finished their journeys by train even then, or they'd have chosen a different site. There isn't any way of getting the station nearer to the current passenger terminal without completely rerouting the railway itself. The idea of a shuttle bus was tried and hardly anybody used it, so it was abandoned. There were frequent scheduled buses from the airport to Darlington and to the Teeside towns, but these have dwindled away too.
I'm not sure if this is true. ISTR having seen a photo of an old passenger air terminal near the station site in the dim and distant past, but can't find any trace of that online now. Published histories suggest the current terminal site, albeit since further rebuilt, has been in use since opening by Princess Margaretha of Sweden in 1966.
The big question for me is what was this structure, seen here as a metal framework in 2000?
That's the earliest it appears clearly on Google Earth historic imagery. It's shown in subsequent images until 2008 but had been removed by 2014.
Was this an abandoned new construction never finished, or the remains of something decommissioned? A freight terminal? admin building? Certainly a good position for a rail served passenger air terminal if one was desired, with apron and taxiway arrangements altered to suit.
So there are three options: do nothing, and wait for the station to fall down (which is what has actually happened), or move the terminal building again, or move the entire trackbed. Since the last two involve very large amounts of public money which would be very hard to justify even if every single airport passenger and worker used the train (because there are so few of them) it is most unlikely anything will happen at all unless the airport closes and is redeveloped for commercial/housing/industrial purposes. The airport is once again in public ownership and the local elected mayor has a lot of political capital invested in this, so...
When the station was built, it looks like the current terminal site had already been chosen and built. Early airport developments rarely took any account of what developers often considered the outdated and soon to be abandoned railway network. By now we should all be zoomng to the airport in our flying cars anyway.