The closest airport is Hannover (HAJ). Often, however, it is much more affordable to fly into Frankfurt am Main (FRA) and to take a train from there.
From the airport in Hannover, take the commuter train (S-Bahn) to the main train station (Hbf). From there, take the train to Goslar/Harz. After arriving in Goslar, go directly to the Bus Terminal, which is located directly at the train station exit. Board either Bus 830 or 831 in the direction of Clausthal-Zellerfeld and exit at the bus station Kronenplatz. During the day, the busses run twice an hour, every 10 or 26 minutes past the hour.
The travel time from Hannover main station to Clausthal takes about 1.5 hours altogether. Train tickets can be purchased at the airport, at the Hannover main station or online under www.bahn.de. They cost approximately € 25 for a one-way ticket. The travel time from Frankfurt to Clausthal takes about 5 hours and a ticket costs approximately € 75 for a one-way ticket.
For further information about train connections and tickets can be found on the websites of the Deutschen Bahn AG (from: Hannover Flughafen/ Frankfurt am Main Flughafen to: Kronenplatz, Clausthal-Zellerfeld).