Fun fact: The Auto-industry created the term "jaywalkers" to shame pedestrians off the streets and into cars. A "Jay" in the early 1900s meant "simpleton", or not as nice way to put it - "hick", for those from the rural parts in the US who were making their way to the cities for the first time. In this case, referring to these people as "too simple" to know traffic laws.
Blue Jays are also not known to be the brightest bird - often being loud, obnoxious, and uncoordinated - so it was a doubly insulting to be referred to as a jaywalker. In retrospect, such a creature might not be the best mascot for your professional baseball team.