Sweep any loose debris or dust from your stairs into a dustpan and discard the dust. Alternatively, use a microfiber dust mop to collect the dust to be discarded later.
Wash the stairs with a commercial wood cleaner. Dilute the cleaner as directed by the manufacturer, then wipe the solution over the wood with a clean cotton cloth. If directed by the manufacturer, rinse the wood cleaner away with a damp cloth. Some cleaners don't require a rinsing process and can be left to dry.
Allow the stairs to dry completely. Rent a handheld buffer designed for use on wood surfaces. These can typically be rented from hardware and home improvement stores -- the store clerks can help you find what you need or direct you to a store that has the appropriate buffer and buffing pads.
Apply a small amount of wood floor polish to the buffing pad. Place the pad against the top step and switch the buffer on. Slowly buff the wood using broad, even strokes -- avoid letting the buffer linger on any one spot. Buff until the wood shines and feels relatively dry to the touch.
Buff each step, working from the top of the staircase down to the bottom. Move the buffer over each step until it comes to a uniform shine; apply more wood floor polish when necessary. Allow the steps to dry completely before walking on them. Drying times vary by manufacturer, so read the label for specific information.