Preheat the oven to 350°F (175°C). Grease and flour a 9-inch square baking pan.
In a medium bowl, whisk together 1 1/2 cups flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and 1/4 cup brown sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
Mix in the mashed bananas and sour cream until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Fold in the chopped walnuts if using.
Pour the batter into the prepared baking pan and spread evenly.
To make the streusel topping, combine 1/2 cup flour, 1/2 cup brown sugar, cinnamon, and salt in a small bowl.
Cut in the cold butter pieces with a pastry cutter or your fingers until the mixture resembles coarse crumbs.
Sprinkle the streusel evenly over the batter in the pan.
Bake for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and let cool in the pan on a wire rack for at least 20 minutes before slicing and serving.