Conventional stairs are constructed by having 'Strings' - the side supports, 'treads' - the board you stand on, and 'risers' - the upright boards between the treads. The treads and risers are housed into the strings in grooves, and the treads are tightened into the grooves by wedges. If you could get to the back of the stairs (which you probably can't) you could hammer the wedge in tighter. The reason it is squeaking is because the timber has dried and shrunk allowing slight movement somewhere.
I doubt if knocking in nails willy-nilly would solve the problem. There is a German product called 'Holzfest' (trans. woodfirm) made by Uhu.
https://translate.google.de/translate?hl=en&sl=de&u=http://www.uhu.com/de/produkte/holzleime/detail/uhu-holzfestholzquellmittel.html%3FcHash%3Ddd82f7b023ac7e58c2064743c7ab966d&prev=search
Which is a thin viscous glue that seeks its way into loose joints and then dries and sets. If you can obtain some, you could try to apply it all around the joints between the strings, treads, and risers and step on and off the tread to allow the liquid to penetrate.
It might work - I hope!