Mix trisodium phosphate with water according to package directions. Clean the high chair with the mixture. Rub the furniture until the dirt begins to lift. Rotate the rag, switching to a clean portion of the rag as it gets soiled.
Dip a piece of 0000 steel wool into the trisodium phosphate solution. Rub the bare wood gently with the steel wool. Press lightly to avoid scratching the wood.
Sand the high chair with fine-grit sandpaper. Remove the top layer of finish from the high chair. Fold a piece of sandpaper in half. Use the corner to work the sandpaper into small crevices. Wipe away sanding dust with a damp rag.
Dip a rag into the wood stain and rub the stain into the wood. Apply two to three coats of stain. Allow the stain to soak into the wood before applying the next coat.
Rub paste wax onto the surface of the high chair. Wax will protect the finish. Apply a thin coat of wax.