Preheat the oven to 325°F (160°C).
In a medium bowl, combine graham cracker crumbs, granulated sugar, and melted butter. Mix until well combined.
Press the graham cracker mixture firmly into the bottom of a 9-inch springform pan to form the crust.
In a large mixing bowl, beat the softened cream cheese until smooth.
Gradually add powdered sugar and vanilla extract, mixing until fully incorporated.
Add the eggs one at a time, mixing well after each addition.
Fold in the sour cream, blackberries, raspberries, cornstarch, and lemon juice until evenly distributed.
Pour the cheesecake batter over the prepared crust in the springform pan.
Bake in the preheated oven for 30 minutes, or until the center is set but slightly jiggly.
Turn off the oven and leave the cheesecake inside for an additional 10 minutes.
Remove from the oven and allow to cool to room temperature.
Refrigerate for at least 4 hours or overnight before serving.