The AA routefinder gives the distance as about 3 miles, with a driving time of 5 minutes. Based on that, the fare is probably not much more than a fiver.
For an accurate quote, phone Station Taxis on 01302 368636.
Alternatively, buses run directly to the Dome from Doncaster Transport Interchange (which incorporates the rail station) every few minutes. These buses will get you there:
55, 55B(Pink Line)
57(Brown Line)
58, 59 (Lime Line)
75(Olive Line)
91
Should be in the region of a fiver for a taxi. The route isn't far, relatively straight forward - dual carriageway all the way. There are black cabs always outside which obviously work on distance plus time spent. you would be better calling a privatehire - try Micks on 01302 820282 or Apple on 01302 342000 they can pick you up and take you for less than a black cab.
If you come out of the station and turn left you will be directly into the new transport interchange where buses are frequent.