Bring a saucepan of salted water to a boil. Add the haricots verts and cook until crisp-tender, 3 to 5 minutes. Drain and run under cold water to stop the cooking.
Puree the yogurt, mayonnaise, lemon juice, chives, basil and anchovies in a blender until smooth. Season with salt and pepper.
Toss the crabmeat, half of the avocado and about 1 tablespoon of the yogurt dressing in a small bowl. Toss the romaine, croutons, haricots verts and the remaining avocado with the remaining dressing in a large bowl.
Divide the romaine salad among plates, top with the tomatoes and place some of the crab mixture in the center.