Tip: Use a vegetable peeler to easily remove strips of the outer skin (zest), leaving the bitter white pith behind.Note: A non-reactive pan-stainless steel, enamel-coated or glass-is necessary when cooking acidic foods, such as lemon, to prevent the food from reacting with the pan. Reactive pans, such as aluminum and cast-iron, can impart an off color and/or off flavor in acidic foods.
Combine water, sugar, lemon zest, lemon juice and cinnamon stick in a large non-reactive saucepan (see Note); bring to a simmer over medium-high heat, stirring often, until the sugar dissolves. Add peach halves. Return to a brisk simmer, cover the pan and simmer, turning the peaches occasionally, until they are tender when pierced with a skewer or paring knife and the skins are loosened, 20 to 25 minutes (depending on the ripeness of the peaches). Transfer the peaches to a plate with a slotted spoon.
Return the liquid to a boil and cook until reduced to about 3/4 cup, 10 to 12 minutes. Remove from the heat, stir in basil and let cool to room temperature, about 40 minutes.
Slip off and discard the peach skins. Place the peaches in a storage container and strain the cooled syrup over them. Cover and chill for at least 4 hours.