Indirectly, the unions helped elect Nixon. Oh, sure, they spent $10,000,000 or more in 1968 dollars to elect Humphrey, and did turn around Michigan and a couple of other northern states. But what they did in Chicago was brutal.
Everyone blames Daley for the police and the situation in Grant Park. In fact, White makes it clear that the police acted properly over the first three days of the convention. The real brutality only occurred after the convention, when they stormed the 15th floor of the hotel, and took out their frustrations on the McCarthy kids, who had had nothing whatever to do with the SDS-organized mayhem outside. That wasn't shown on TV.
A telephone strike meant that the video from outside couldn't be transmitted live for broadcast; a taxi strike, coupled with Mayor Daley's refusal to let the TV trucks park on the sidewalk outside the convention hall, meant that the tape couldn't even be transported reliably.
Today, the riot would have been televised live, and the nomination and acceptance would have been televised later. As it transpired, Humphrey's acceptance speech was broadcast at the same time as the riot videotape, helping to give the impression that Humphrey was being installed as nominee by bayonets and tear gas.
And the unions were the reason for that.
Dems and ballot initiative voters, take note.