Peel and finely chop the onion. Put it with the olive oil in a soup pot over medium-high heat. Sauté, stirring occasionally, until the onion turns a rich golden color, about 5 minutes.
While the onion is sautéing, cut the zucchini into ½-inch dice. Peel the carrots and cut into ½-inch dice. Snap or cut off the ends of the green beans and cut into ½-inch lengths. Peel the potatoes and cut into ½-inch dice.
When the onion is done, add the vegetables to the pot. Season with salt and pepper and sauté for about 5 minutes.
Add the broth and bring to a boil. Lower the heat so that the broth simmers, then cover the pot and cook for 15 minutes. Use a slotted spoon to lift out about half of the vegetables and purée in a food mill or food processor. Return the puréed vegetables to the pot. Bring the soup back to a simmer and add the barley. Cook, covered, for 20 minutes.
Coarsely shred the basil with a knife and add it to the soup. Cook until the barley is tender, about 10 minutes. Serve at once.