The big problem Quinn has here is that while Cohen is technically his "running mate," in the sense that they will be listed together on the general election ballot in the same way that we vote for president and vice president, Quinn did not pick him. Candidates for governor and lieutenant governor in Illinois are nominated in separate primaries, but then run together as a ticket.
Quinn won his own primary with just over 50% in a two-way race. Meanwhile, Cohen was nominated with a 26% plurality in a field of six candidates -- and Quinn has an obvious political problem on his hands.
This system of nomination infamously led to a serious problem in Illinois elections back in 1986, when a LaRouche activist won the Democratic nomination for lieutenant governor. The Democratic nominee for governor, former Sen. Adlai Stevenson III, then took the extraordinary step of abdicating his own nomination by the Democratic Party, and instead forming a third-party ticket in order to run in the general election with a mainstream Democrat at his running mate.
In the end, incumbent Republican Gov. Jim Thompson won re-election with 53% of the vote, to 40% for Stevenson's "Solidarity Party" ticket.