Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking dish.
Cream butter and sugar until light. Add eggs, mashed bananas, and vanilla. Mix until smooth.
In another bowl, combine flour, baking powder, baking soda, and salt. Gradually mix dry ingredients into the wet mixture, alternating with sour cream.
For the filling, beat cream cheese, sugar, egg, and vanilla until creamy.
Pour half the batter into the dish, spread the cream cheese filling evenly, and layer with the remaining batter.
To make the crumb topping, mix brown sugar, flour, and cinnamon. Cut in butter until crumbly. Sprinkle evenly over the batter.
Bake for 45 minutes or until a toothpick inserted into the cake comes out clean.
Let the cake cool slightly before slicing and serving.