Wash basil leaves carefully in cold water to remove any sand that may be clinging to leaves. Drain and pat dry with paper towels or a clean old tea towel, as basil may stain. Squeeze juice from lemon.
Place basil, 1 tablespoon lemon juice, pine nuts, garlic, salt and pepper in a blender or food processor. Whirl, using an on-and-off motion, until fairly smooth, scraping down sides of bowl frequently. While machine is running, gradually add olive oil drop by drop at first, then in a thin steady stream, until all oil is added. Taste and swirl in more lemon juice, 1 tablespoon at a time, if needed.