Place the potatoes in a medium saucepan with enough water to cover. Bring to a boil, lower the heat and simmer until fork-tender, about 15 minutes; drain.
Meanwhile, in a large pot, heat the olive oil over medium-low heat. Add the garlic and cook, stirring, until golden. Add the broccolini, season with salt and pepper, cover and cook, stirring occasionally, until crisp-tender, about 3 minutes. Transfer the broccolini to a bowl.
Add the clams, wine and 1 cup water to the pot. Cover and cook over high heat, stirring occasionally and transferring the clams as they open to 4 soup bowls, for about 5 minutes. Slice the potatoes and add to the soup bowls.
Add the tomatoes and beans to the pot and bring to a boil. Return the broccolini to the pot and heat through. Transfer the broccolini to the soup bowls, then ladle the broth and beans on top. Top with the basil.