In a small bowl, mix together the goat cheese and basil. Season with salt and pepper, to taste. Transfer the contents of the bowl to a pastry bag fitted with a round 1/4 inch decorating tip.
Gently pull the dates apart and stuff with cheese mixture. Close the dates around the filling. Wrap a piece of prosciutto around each date and secure with a toothpick.
Heat broiler to high and broil until cheese bubbles, about 3 minutes. Serve warm.