Preheat the oven to 350°F (175°C) and grease a 9x13-inch baking pan.
Cream butter and sugar until fluffy. Add eggs, mashed bananas, and vanilla extract. Beat until smooth.
Combine flour, baking powder, baking soda, and salt. Gradually mix dry ingredients into the wet, alternating with sour cream.
Pour the batter into the prepared pan and spread evenly.
For the crumb topping, mix brown sugar, flour, and cinnamon. Cut in butter until it forms coarse crumbs. Sprinkle over the batter.
Bake for 50 minutes, or until a toothpick inserted in the center comes out clean.
For the caramel drizzle, heat sugar in a saucepan until melted and amber in color. Add cream, butter, and a pinch of salt. Stir until smooth.
Drizzle warm caramel over the cooled cake before serving.