My stairs are old and squeak and I would like to screw them down and plug the holes. My question is what length screw should I go with? They are about an inch thick. The stringers are old as it's antique so I guess I was concerned not to go too long, I'm not sure if that can affect the strength of the stringer.
Drill pilot holes (with a slightly smaller diameter than the screws' root diameter). Use screws that are between 2 and 4 inches long. Deck screws are available in this size range, and are used for this purpose on deck stairs.
Drilling pilot holes will help prevent splitting the wood of the stringers.
Stringers are often made of dimensional lumber, so they are often only about 4" thick at their thinnest points.