The best material to use is a layer of elastomeric roofing membrane,
more commonly known as ice and water barrier. Its use extends not only
to roofs, where it is used one everything from flashing to chimneys,
but also to carpentry and framing, where it is used around windows and
under mudsills for the same reason as your stairs.
This material is a fairly thick rubbery membrane, imperivous to water,
with an extremely sticky coating on one side. Once you stick it down,
it is nearly impossible to reposition. You should lay the membrane
down in a strip where the stringers will fall.