Cut and arrange the fruit: Arrange four 9-inch serving plates on a work surface. Place the strawberries, hulled-tops down, on a cutting board. Cut each strawberry into thin slices and lay them in a single layer on the plates. Save any strawberry scraps or unevenly cut pieces in a medium bowl. Intersperse the raspberry halves in the gaps on the plates. Reserve any raspberries that are too soft in the bowl with the strawberry scraps. Refrigerate the plates of berries.
Make the dressing: Press down on the berry scraps with the back of a spoon as if you were muddling fruit for a drink. Whisk together the cranberry juice, olive oil, honey and ginger in a small bowl along with the zest and juice of the lime. Add to the berry scraps and taste for seasoning.
Serve: Remove the plates from the fridge, then drizzle the dressing over the fruit and sprinkle with a light, even layer of the sugar. Garnish with the basil and serve.