Place a medium pot over medium heat with the cranberries, orange juice, apple cider (hold back about 2 tablespoons), sugar, cinnamon and cayenne pepper. Season with salt and pepper.
Bring liquid up to a bubble then reduce the heat to medium and cook until the cranberries start to break apart and the mixture becomes saucy, about 5 minutes.
In a small bowl, stir together cornstarch and reserved apple cider, and stir the mixture into the simmering cranberries. Add the apples and cook until tender, and the mixture thickens, 3-4 minutes. Stir in the basil and let cool to room temperature. Serve at room temperature.