Place potatoes in a large saucepan; cover with water. Bring to a boil. Reduce heat; simmer 20 minutes or until tender. Remove potatoes with a slotted spoon; place on a cutting board. Let cool to room temperature. Add beans to simmering potato cooking water; cook 3 minutes or until crisp-tender. Drain; let cool slightly. Cut potatoes into quarters.
Combine mayonnaise and next 7 ingredients (through garlic) in a large bowl, stirring with a whisk. Add potatoes, beans, onion, and basil; toss gently to combine.