The closest international airport is Duesseldorf (DUS). From DUS trains are operated by Deutsche Bahn, it takes about 90 min to reach Aachen main station (Hauptbahnhof). There is a direct train ones per hour, costs are about 24 Euro. The last train to reach Aachen leaves about 22:30. See here for the train schedule. Note that the trains leave from "Duesseldorf Airport train station" which is reached by a 5 min ride with the sky train, running every 3-5 min. Follow signs for "sky train" at the aiport (easy to find).
Alternatively one can fly to Brussels airport Zaventem. In particular from Geneva this might be cheaper. From Brussels airport take a train to Brussels Nord or Midi and from there a train to Aachen. Overall it takes about 90 min. The connection is hourly, Deutsche Bahn ICE trains (about 54 Euro) and Thalys (special tickets, to be bought directly from Thalys) alternate. Check for the latest train, they stop running early.
For those enjoying a train ride, Aachen can also be reached by train. Trains from Brussels all go through/to Aachen. Other connections require a change in Cologne. Three times per day, a direct high speed train from Frankfurt Airports goes to Aachen.
Connections by car are very comfortable since three interstates meet in Aachen, coming from Cologne, Amsterdam and Brussels.
