Campaigning in South Carolina, Hillary Clinton on Wednesday proposed a new tax credit for businesses that hire and train apprentices, an effort to increase employment of young adults. These programs draw bipartisan support but have proven a tough sell in the private sector.

“This is not just about helping individuals,” she said at Trident Technical College in North Charleston, S.C., home to one of these programs. “This is about building our economy from the bottom up, with people who have the skills and the drive and the work...