1. In a small bowl or measuring cup, whisk together the orange juice, sherry or red wine vinegar, salt and pepper, and oils. Taste and adjust the acidity, adding a little more vinegar or orange juice if desired.
2. Toss the beets with 1 tablespoon of the dressing. Toss the arugula with 3 tablespoons of the remaining dressing and arrange on a platter or in a wide bowl. Arrange the beets and oranges on top of the arugula in an alternating pattern and drizzle on the remaining dressing. Top with the basil and walnuts, and serve.