Place sour cream, lime juice, garlic, 4 cups cucumbers, 1/4 cup each of cilantro and basil, and 1 minced jalapeño in a blender; process until smooth, about 45 seconds. Transfer puréed mixture to a bowl; stir in sparkling water, salt, pepper, and remaining cucumbers and jalapeño.
Place bowl with gazpacho into a larger bowl filled with ice. Let stand, stirring often, until gazpacho is cold, about 20 minutes. (Or place in the refrigerator 1 hour.)
Place crabmeat, oil, and remaining 1/4 cup each of cilantro and basil in a bowl; stir gently. Divide gazpacho evenly among 6 bowls; top with lump crabmeat mixture, and serve immediately.