Make the dressing: Whisk together lime juice and ginger in a small bowl. Gradually whisk in oil. Season with salt and pepper.
Make the salad: Slice or shave carrots, fennel, radish, rutabaga, and turnips, preferably on a mandoline, into very thin rounds over a large bowl; toss. Add vinaigrette, jalapeno, cilantro, and mint, and toss to combine. Season with salt. Gently fold in crab. Garnish with cilantro and mint.