Preheat oven to 375 degrees F (190 degrees C). Grease and flour a 9x13-inch pan.
Beat sugar and butter together in a bowl using an electric mixer until creamy and smooth; mix in milk, eggs, and vanilla extract.
Sift flour, baking powder, baking soda, cream of tartar, and salt together in a bowl; stir flour mixture into butter mixture until just combined. Fold in bananas and walnuts. Pour batter into prepared pan.
Bake into the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 30 minutes.