Combine melted butter, brown sugar, granulated sugar, cinnamon, ginger, and flour for the crumb topping until it resembles wet sand; chill in the refrigerator.
Preheat oven to 350°F and grease a standard cake pan.
Whisk flour, baking powder, baking soda, and salt in a small bowl.
Cream butter and sugar in a processor until light and fluffy.
Beat in the eggs and vanilla extract until incorporated.
Fold in half of the dry ingredients, followed by the sour cream.
Fold in the remaining dry ingredients and gently add the blackberries.
Spread batter into the prepared pan and top generously with the chilled crumb mixture.
Bake for 50 minutes or until a toothpick comes out clean from the center.
Cool before serving.