There has frequently been discussion of a direct bus link from Sheffield. However I expect most passengers from the airport either use the parking facilities, get a lift off a mate or share a taxi. There used to be a dedicated fast bus to Doncaster Interchange (next to the station) but this was withdrawn due to lack of use. The normal service buses are not much slower.
As you will know, there are many flights from Eastern Europe to Doncaster. I have used these a few times and on anecdotal evidence I have never seen more than fifteen off each flight transfer to the bus. Many of these do seem to be going to Sheffield, however, and better information on the bus shelter on ways to get there may be helpful: for example if the traveller is not in a rush, a First day ticket is much cheaper than the train.