Set the stair post on the floor in the desired location. Make sure the post is square and in line with the stair railing. Trace around the base of the post with a pencil. Remove the post.
Place an L-bracket on one side of the outline on the floor. Position the bracket so the part laying flat on the floor is inside the outline. Line up the inside of the upright part of the bracket with the outline mark on the floor. Screw the bracket to the floor with 2-inch wood screws. Install a screw in each hole, and drive the screws with a drill and driver bit. Repeat this process for the other three brackets. When installed, the upright parts of the brackets should form a four-sided bracket that's the same size as the bottom of the post.
Set the post into the brackets. Push down on the post to fully seat it. Make sure the post is plumb by checking two adjacent sides with a level. Tap shims under the low sides of the post to make it plumb, if necessary.
Drive 2-inch wood screws through the holes in the upright parts of the L-brackets, then into the stair post. Insert screws in all of the holes to secure the post.
Measure and cut four pieces of baseboard trim to fit around the base of the post. Miter the ends of the baseboard trim pieces at a 45-degree angle. Assemble the pieces of trim around the base of the post, then secure them into place with trim nails. Nail the trim near the ends to avoid hitting the L-brackets.