A series of articles appeared in the Appleton Post-Crescent in 1940: "Appleton Streets How They Were Named." The February 20th entry for State St. ssys this street was named for the state of Wisconsin.
The entire 300 block of S. State St. from W. Eighth St. to W. Seventh St. is taken up by St. Mary's Catholic Church parish buildings: a parish hall, the church, and Catholic Central Elementary School.
A portion of Appleton's electric trolley line ran up and down State St. from Prospect to College Ave. A video of the last run of the trolley in 1930 can be seen here.
From Morrow and Gillett's Appleton city directory, for 1889-90:
STATE, 1000 Second, north to Second av, fourth west of Appleton, numbers 340 to 1000.
In 1925 a new system of numbering structures in the city of Appleton was adopted. In this directory both the new and old street numbers are given. In addition, College Ave. was designated the dividing line between north and south, and Oneida St. was the east-west divider.
From Wright's Appleton city directory 1925:
STATE S - From 601 W College av s to Fox River, 5 w S Oneida