The nearest airport is in Hanover (HAJ). However, it is often much cheaper to fly to Frankfurt am Main (FRA) and take the train from there.
From Hanover airport, take the S-Bahn to the main train station (Hbf). From there, take the train to Goslar/Harz. When you arrive in Goslar, go directly to the bus terminal, which is located directly at the station exit. Take bus number 830 or 831 to Clausthal-Zellerfeld/ Kronenplatz station. During the day, the bus runs twice an hour; always at 10 or 26 minutes past the hour.
The total travel time from Hanover main station to Clausthal is about 1.5 hours. Train tickets can be purchased at the airport, at Hanover main station or online at www.bahn.de can be purchased. They cost around 25 euros for a one-way ticket. From Frankfurt to Clausthal it takes around 5 hours and a ticket costs around 75 euros one way.
Further information on train connections and tickets can be found on the Deutsche Bahn AG website (from: Hanover Airport/ Frankfurt am Main Airport to: Clausthal-Zellerfeld/ Kronenplatz).