Preheat an oven to 275 degrees F (135 degrees C). Spread the walnuts onto a baking sheet, and toast until the nuts start to turn golden brown and become fragrant, about 30 minutes. Watch the nuts carefully as they bake, they burn quickly. Once toasted, set the nuts aside to cool to room temperature.

Place the potatoes into a large pot, and cover with salted water. Bring to a boil over high heat; reduce heat to medium-low, cover, and simmer until tender, about 15 minutes. Rinse under cold water, and place in large bowl. Refrigerate until cold.

Combine apples, onion, cheese, and walnuts with cooled potatoes. Whisk together the olive oil, vinegar, basil, salt, and pepper in a small bowl. Pour the dressing over the salad, and toss to coat. Serve chilled.