Look over the newel post and try to find the pilot hole in the middle of the bottom of the newel post. The pilot hole is a small hole that is used to guide screws. Avoid any wood plugs.
Drill a counterbore, or a larger hole where the pilot hole is located, into the wood with a 3/4-inch drill bit. The hole should be 3/4 inches deep and angled up.
Drill a pilot hole through the middle of the counterbore you previously drilled and into the base of the stair. Widen the pilot hole so that it is 5/16-inch wide through the newel post, being careful to not go too deep, as the wood could split. Take a 4-inch-long screw and put a 3/4-inch diameter washer on the top of the screw. Place the screw in the pilot hole and tighten with a screwdriver.
Fill the counterbore with wood glue and slide in a 3/4-inch diameter wooden dowel. Wipe up any wood glue that spills out of the hole.
Paint the area you repaired after the glue has dried with paint that matches the newel post's existing finish. If the wood is natural, you can use a stain instead.