This term for crossing the street in the middle of the block is U.S. slang dating to the early years of the 20th century. It comes from the sense of jay meaning a stupid person, a rube.
The earliest known use has the term in a slightly different sense, that of someone who is walking aimlessly. From the Washington Post on 7 May 1911:
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Kansas City used to consider itself a town of jay walkers. That is another line in which New York deserves the discredit of being at the front of the procession. A typical Manhattan would be run over and trampled on the sidewalk if he tried to walk on State street in Chicago as he walks on Broadway, New York. He has never heard of the prehistoric principle of keeping to the right—he ambles all over the sidewalk.
Copyright 1997-2016, by David Wilton