In the bowl of a stand mixer fitted with a paddle attachment, beat the cream cheese and sugar on high speed until smooth.
Add the vanilla extract, then mix in the eggs one at a time, beating just until incorporated.
Add the sour cream and mix until smooth. Gently fold in the mashed bananas.
Pour the cheesecake batter into the prepared crust.
Wrap the outside of the springform pan tightly with two layers of foil to prevent water from seeping in.
Place the springform pan into a large roasting pan and carefully pour boiling water into the roasting pan until it reaches halfway up the sides of the springform pan.
Bake for 50-60 minutes, or until the edges are set and the center is slightly wobbly.
Turn off the oven, crack the oven door open, and allow the cheesecake to cool for 1 hour inside the oven.
Remove from the water bath and let the cheesecake cool completely to room temperature.