In what ways did the peasants affect the plays written during Shakespeare's life?
True peasants? Uneducated farm laborers, living in small villages? Almost not at all. They are minor characters in some of his plays, but not very realistically.
If you mean urban working classes, they shaped the plays somewhat more. Shakespeare's audience included masses of fairly poor working people, and they had limited education. He often crafted plays to have something for everyone: lines for the upper classes, and lines (and action) for the working class.