Place the chopped peaches in a bowl or mason jar and set aside.
In a saucepan, combine the sugar and 2 cups of water. Stir to combine. Turn the heat to medium high and boil until the syrup becomes thicker but is not yet starting to turn color, 3 to 5 minutes. Remove from the heat and pour the syrup over the chopped peaches. Allow to sit at room temperature for 1 hour (or you can chill in the fridge).
To serve, spoon the peach syrup over the vanilla ice cream and sprinkle with the basil.