Karlsruhe is located close to Frankfurt International Airport. The best way to reach Karlsruhe from Frankfurt Airport is by train. It only takes about one hour and trains depart directly from the airport.
Alternatively, Karlsruhe/Baden-Baden Airport is a smaller Airport even closer to Karlsruhe. You can reach the city by bus and train within one hour.
You can find more information about the exact arrival and departure times of trains and busses from the Deutsche Bahn AG.