Stir together mayonnaise and next 5 ingredients in a small bowl until blended; cover and chill.
Bring potatoes and water to cover to a boil over medium-high heat; cook 10 to 15 minutes or until potatoes are fork-tender. Drain potatoes, and let stand 25 minutes at room temperature or until cool. Cut into 1/4-inch-thick slices.
Cook green beans in boiling water to cover 3 to 4 minutes or until crisp-tender; drain, and plunge into ice water to stop the cooking process. Drain and pat dry.
Layer half of potato slices in a large glass bowl; sprinkle evenly with 1/2 teaspoon salt. Layer half each of green beans and onions over potatoes; spoon half of dressing evenly over top. Repeat layers, ending with dressing. Cover and chill at least 4 hours or up to 24 hours.
*2 (12-ounce) bags fresh ready-to-cook trimmed green beans may be substituted.