Demolish the existing staircase by prying off each tread with the back of a hammer. The treads are the flat pieces of wood you step on. These will probably be secured with a few nails. Remove the rises in the same way; these are the vertical boards that make the back of the step. You will be left with the two stringers. The stringer is the tall triangle shape of wood, with a row of jagged teeth onto which the stairs were affixed. Retain one stringer to use as a template.
Trace the stringer onto three sheets of wood with a pencil. Cut out the shapes with a jigsaw.
Stand the stringers up where the old ones were removed from. They should be evenly spaced. Secure the front inside corners of the stringers with fence-post bases. The bases should be installed on the inside of where the bottom step will be on the outside stringer and on either side of the middle stringer.
Drill two holes in the basement floor with a commercial hammer drill in each location of the bases. The distance between the holes should be according to the holes in the base bracket of the fence-post base. Install a small block of lumber into the fence-post base with a screwdriver.
Attach the fence-post to the stringer by installing a brace between the stringer and the block of lumber in the base.
Start laying the treads at the bottom of the staircase. Nail the first tread into place. Place the first rise against the stringers on the first step level. Nail this in place as well. Continue to attach the treads and rises in this manner until you reach the top.
Screw a handrail connector to the basement beam next to the staircase. Fix the handrail onto the connector.