Preheat oven to 350°.
Combine first 4 ingredients in a medium saucepan; cook 10 minutes over medium heat or until apples are soft. Place apple mixture in an 8-inch pie plate. Sprinkle blackberries over apple mixture.
Lightly spoon flour into dry measuring cups; level with a knife. Place flour in a medium bowl; cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Add granulated sugar and ginger; toss well. Sprinkle flour mixture over fruit. Bake at 350° for 30 minutes or until bubbly.
Broil crumble 1 minute or until lightly browned.