Dampen a clean cloth with water. Wipe the scratched wooden stairs with the slightly moist cloth to collect any accumulated soil and grime.
Wipe the moisture off the stairs using a towel. Thoroughly dry the scratched wood.
Select a colored putty stick that very closely, if not perfectly, blends with the stairs. Putty sticks come in numerous hues to match any wood grain.
Rub the colored stick over the scratches until they are overflowing with putty wax. Apply the putty parallel to the wood's grain lines.
Skim the excess putty off the stairs using a rubber spatula. Leave the scratches fully filled with putty, of course.
Allow the putty-filled scratches to air dry.