Combine first 8 ingredients in a food processor; process until smooth.
Bring water to a boil in a large saucepan. Add green beans, and cook 4 minutes or until tender. Remove beans from pan with a slotted spoon, and place in a large bowl. Add potatoes to pan, and cook 6 minutes or until tender. Drain. Add pesto and potatoes to beans, and toss to coat.