I am installing a toilet in new construction where the flange is set at the subfloor level and slightly below (about 3/8") the level of the hardwood flooring. Does the flange have to be raised to the level of the finished floor? If so, how?

It doesn't have to be, but it helps. You can install a wax ring (with flange), then add a wax ring without a flange on top. This will make up the space difference.

XenoSapien

If you will look under a toilet bowl you will see a horn about 3 or 4 inches long. This horn extends below the floor line and guides the discharge into the closet bend past the floor and the flange. Your flange should be OK but if you're concerned use a wax ring with a funnel. Good luck, tom

3/8 inch isn't too bad. You can buy an extra thick wax ring with the funnel Tom was talking about. Did you anchor your flange to the subfloor?