Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.
Place red bell pepper quarters skin-side up onto a broiler pan or baking sheet.
Cook under the preheated broiler until skins have blackened. Transfer bell peppers to a brown paper bag or a covered bowl and allow to steam until skins loosen, about 10 minutes.
Peel skins off and discard. Coarsely chop peppers and place into a large saucepan along with chicken broth and potato. Bring to a boil, then add half-and-half, salt, garlic powder, basil, and cayenne pepper. Return to a boil, reduce heat to medium, and simmer until potatoes have softened, about 45 minutes.
Purée soup in batches in a blender until smooth. Pour puréed soup into another saucepan and stir in crabmeat. Heat over medium heat until warmed through, about 5 minutes.