Bring 1 in. water to a boil in a saucepan. Set whole potatoes in a steamer basket and steam in pan, covered, until tender, 15 to 20 minutes. Cool in ice water, then pat dry.
Whisk yogurt, mayonnaise, dill, vinegar, salt, and pepper in a small bowl to combine.
Quarter potatoes and put in a large bowl. Add onion, cucumber, and half the dressing; gently stir to coat. Add more dressing if you like, or save to use as a dip.
Make ahead: Up to 2 days through step Chill potatoes and dressing separately and slice cucumber just before serving.
Note: Nutritional analysis is per 1-cup serving using half the dressing.