The Museum itself is located very close to the Bochum main train station. You may simply take any Deutsche Bahn train which stops in Bochum. From the main train station, you can take the underground train (U35) to get off at the Bergbau-Museum station.
The nearest airport around Bochum is Düsseldorf International Airport (DUS). From there you may use Deutsche Bahn trains to reach Bochum. The journey will take around 45 minutes