Generously grease a fluted tube pan (such as a Bundt(R)).
Stir vegetable oil, eggs, and sour cream in a large bowl until thoroughly combined; stir in cake mix, instant vanilla pudding mix, and basil until batter is smooth.
Pour batter into prepared pan.
Bake until the cake is lightly browned at the edges and a toothpick inserted into the thickest part of the cake comes out clean, 40 to 45 minutes.
Let cake cool in pan for 20 minutes before inverting to turn cake out onto rack to cool completely.
Stir balsamic vinegar with sugar in a saucepan and place over medium heat. Bring mixture to a boil, reduce heat to low, and simmer until vinegar syrup is slightly thickened, about 5 minutes. Remove from heat and cool syrup to room temperature.
Just before serving, stir strawberries with balsamic syrup in a bowl; spoon berries and syrup over the cake. Sprinkle cake with fresh mint.
