The rapid ascent of Sen. Bernie Sanders in the Democratic primaries—despite a weak showing in Saturday’s South Carolina vote—confronts American business with the once unthinkable: The most powerful policy maker in the world could soon be a strident, lifelong critic of capitalism and big business.

Current and former business leaders and analysts have greeted the prospect with a mix of concern, skepticism and disbelief. Most doubt the Vermont independent, who calls himself a democratic socialist and caucuses with Democrats,...