The closest airports are Carcassonne and Perpignan, but Toulouse-Blagnac offers many more flights. From Toulouse to Les Labadous is ca. 2 hours by car, from Carcassonne it’s an hour.

Public transportation will get you from the airport to Couiza, a small town in the Aude valley, some 7 km from Les Labadous. While you can reach Les Labadous by taxi, it is certainly advantageous to rent a car, also to reach shops, restaurants, banks, etc.

If you arrive by car, do not to rely on your navigation system for the last 10 km of road when you drive to Les Labadous. (Else you may be directed to a rough road that requires a four-wheel drive.)

Instead:

From Carcassonne take D118 to Limoux. Continue through Limoux in the direction of Quillan, still on D118. When you arrive in Couiza, take the second left after the traffic lights, and head up D52 towards Rennes-Le-Chateau.
From Quillan take D118 towards Carcassonne. Stay on D118 until you arrive in Couiza. In the village, do not miss the right turn to D52 towards Rennes-Le-Chateau.
THEN…
Once on D52, follow this winding road towards Rennes-le-Château. After 2.5 km you will come to a 3-way fork. The main road bends to the right and continues up to Rennes. Instead, you take the road straight ahead, ignoring the no-entry sign. Descend a short distance, take the first right. A blue sign indicates Les Labadous, which is awaiting you at the bottom of the slope, to the right of the road, nestled among large trees.