Sprinkle baking soda over any greasy areas or oil stains on the wooden stairs. Leave the baking soda in place for 15 minutes to absorb the grease.
Sweep the baking soda, as well as any other dirt and loose debris, off of the wooden stairs with a broom and dustpan. Begin sweeping at the top step and work your way down to the bottom.
Fill a bucket with 1 gallon of hot water. Add two cap-fulls of oil-based wood soap.
Soak a clean cloth in the soap solution. Wring it out to remove excess liquid.
Scrub the surface of the wooden stairs to remove any remaining stains, oil and dirt. Rinse the cloth with water as needed. Add more of the diluted soap solution and continue to scrub the stairway until you've washed all of the steps.
Dry the surface of your wooden staircase with another cloth.
Tackle any stubborn stains that remain on the wood steps with a pea-sized amount of toothpaste on a nylon mesh sponge. Scrub the stain with the toothpaste to loosen it up. Rinse the wood surface with a damp cloth then wipe it dry.