A good selection both in the resort itself and in surrounding villages.
The best restaurant in town is Pierre Alexis, which has a magnificent collection of wines lining the walls of the establishment try one of the Barbarescos. Cadran Solaireis a high quality establishment much frequented by discerning Italian visitors. La Terraza and Mont Frety do the best pasta and pizzas and the Vieux Pommier is the place for fondues.
Out of town at Entreves, La Maison de Filippo is famous for its blow-out menus 30-odd courses for those who have the stamina. At Val Ferret Chalet Proment, alias Florianos, is in a wonderful remote position and has a great rustic ambience.
One of the better choices in the alps.
Since many Italians enjoy coming to the mountains but not actually skiing, Courmayeur is actually well geared to cater for non-skiers. The old village itself is very attractive and its streets are lined with pretty shops selling everything from wonderful fresh vegetables to designer clothes. The bars and cafes mostly remain fairly lively during the middle of the day and so the place feels far from deserted.
It is very easy for non-skiers to go up to Plan Checrouit to meet skiing friends and family for lunch at one of the restaurants there. After lunch the non-skiers can happily laze in the sunshine outside the restaurant until the end of the afternoon. Alternatively, there are some pleasant walks along the valley floor, not to mention excursions to Aosta (famous for its market) and the cable car ride up to Punta Helbronner. Day-trips to Chamonix, on the French side of the Mont Blanc tunnel, are an option.