Place potatoes into a large pot and cover with lightly salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 10 to 15 minutes; drain.
Whisk vinegar, olive oil, mustard, basil, salt, and pepper together in a large bowl; add the potatoes and onion. Toss gently to coat. Let stand until cool, about 30 minutes.
Fold blue cheese and chives into potato salad until blended.