Add the parsley, basil, chives, and garlic to a food processor and pulse until finely chopped.
Add the mayonnaise, sour cream, lemon juice, and anchovy paste to the food processor.
Blend on high for 30-45 seconds until the mixture is completely smooth and bright green.
Season with salt and pepper, then taste and adjust seasonings as needed.
Transfer to a serving bowl, cover, and refrigerate for at least 1 hour before serving to allow flavors to meld.
Serve chilled with fresh vegetables, crackers, or chips.