‘Juices’ is a kind of slangy term that means ‘a person’s vitality or creative faculties’. The creative faculties are of course what is referenced in the expression mentioned in the question linked to by jwpat7 in the comment above; in this case, it’s the other meaning we’re looking for.
Since a person’s vitality is seen here as a kind of ‘juice’ (originally most likely referring on some level to bodily fluids: especially blood was, in Mediaeval times, often considered to be a source of energy, health, and vitality in humans), it makes sense that it should be set ‘flowing’, since that’s what liquids do.
This meaning is extended from the meaning the OED has as sense 2:
The fluid part or moisture of an animal body or substance; now usually in pl. the various liquid constituents of the body, the bodily ‘humours’
Obviously, if you think of it as being ultimately just a kind of euphemism for blood (and sweat and bile and all the other humours), getting it to flow, and flow faster especially, would be a sign of physical exertion or excitement: when we run, our blood actually flows faster, and our pulse increases.
The phrase is often also used in a sexual manner, where the reference is to the various kinds of sex-related bodily fluids that are released when people become aroused. Whether the sexual meaning came first or the exertion meaning came first (or whether they are really to be separated at all), I do not know; I cannot find any quotes for that. But it is clear that they are both in use.
In the quote from your NYT article, the meaning is definitely that of physical exertion–based excitement. He is comparing the ‘rush’ he gets from driving to a kind of race where he’s not going to let anyone pass him and be faster than him.
According to this Ngram,
the phrase first appeared in books in 1935, disappeared until 1949, and experienced an early spike in usage around 1951. A spike reaching a slightly lower peak occurred in 1967, followed by intermittent but generally up-trending popularity until reaching another high point in 1985. A slight drop-off then was followed by a sharp increase from the early 90s up to a historical high in 2000, the last year of Google's Ngram-available data.
It seems the first use noted was in The American Spectator, Volume 3, Issue 34:
The editorial bears the heading, "McDonald and Lenin," and he Times
puts itself on the wrong side of history thus: "If the British people
regard the advent of Labor Socialists without alarm, on reason is that
they know that they have to deal with a Labor government soberly aware
of its limitations and its responsibilities...The example of Russia is
a virtual guarantee against any tendency on the part..."
Whether the phrase originated with the author of this piece is not evident.
This version of the same Ngram of the phrase “on the wrong side of history” with a "smoothing" of 50, shows the general trend over time from 1800 to 2000.
Best Answer
It is sloppy English. In your context the phrase would be:
Do you want to have a hand in this?