Stir over medium heat until sugar dissolves and syrup comes to simmer. Using slotted spoon, transfer berries to small bowl. Add half of apple slices to simmering syrup and cook until tender and just translucent, stirring occasionally, about 12 minutes. Using slotted spoon, transfer apples to sieve set over large bowl. Press apples to extract excess syrup.
Transfer apples to another bowl.
Repeat cooking and draining with remaining apples. Return berries and drained syrup to same saucepan. Transfer boysenberry sauce to small bowl. Cover and refrigerate. Crumble cake onto large baking sheet.