Preheat the oven to 325°F (163°C) and grease a 9-inch springform pan.
Combine graham cracker crumbs, 1/4 cup sugar, and melted butter in a bowl. Press firmly into the bottom and 1 inch up the sides of the springform pan. Bake for 10 minutes and let cool.
In a large mixing bowl, beat the cream cheese and remaining 1 cup of sugar on medium speed until smooth and aerated.
Incorporate the mashed bananas, sour cream, lemon juice, and vanilla extract, mixing until fully combined.
Add eggs one at a time, mixing on low speed after each addition just until the yolk disappears to avoid over-incorporating air.
Wrap the exterior of the springform pan in three layers of heavy-duty aluminum foil to prevent water leakage.
Pour the batter over the cooled crust and place the pan inside a larger roasting pan. Fill the roasting pan with boiling water until it reaches halfway up the sides of the springform pan.
Bake for 65 to 75 minutes, or until the edges are set and the center has a slight jiggle.
Turn off the oven and crack the door open slightly. Allow the cheesecake to cool in the water bath for 60 minutes.
Remove from the water bath, discard foil, and refrigerate for at least 8 hours or overnight before serving.