Scrub potatoes and cut into 1/2-inch chunks.
In a 4- to 5-quart pan over high heat, combine potatoes, carrots, and broth. Cover, bring to a boil, and cook until potatoes and carrots are tender when pierced, 10 to 12 minutes.
Meanwhile, remove any bits of shell from crab.
Add buttermilk to pan and stir until hot, 2 to 3 minutes.
Add crab, mix gently, and ladle soup into wide bowls. Add cucumber equally to portions. Season to taste with salt and pepper.