The real reason for the New Left opposition to eugenics (beyond
strategic electoral demographics) has nothing to do with preventing the
state from imposing low fertility upon the lower orders (after all, the
politically correct see no limit to the state's right to control
people, and the Left care nothing for the freedom of the lower orders).
The modern ruling class opposition to eugenics has almost-everything to do with preventing the state from
enforcing high fertility upon the higher orders - ie. the PC elite themselves.
In other words, eugenics is about two things - increasing the differential representation of beneficial genes in the gene pool, and reducing or eliminating the proportion of deleterious genes in the gene pool; and in terms of human fertility this implies higher fertility for the most able and best-behaved, as well as lower fertility for the least able and worst-behaved.
It is my contention that the modern Leftist elites are not really bothered about the fertility of other people; they don't talk or write about this topic and don't lose sleep over it (although they probably should); and they would naturally regard themselves and people like themselves as having the highest aptitudes and best attitudes; however, they are also extremely concerned with controlling, delaying and in general suppressing their own fertility (and duties and responsibilities).
Therefore they dishonestly deny that intelligence and personality are substantially heritable - because this gets them off the hook!
Anything which would tend to pressure the modern elites to focus on marrying much younger, staying married, and quickly start having children, up to three or more...
Well... anything of that kind is regarded with fear and loathing - whether it be eugenics or traditional religion.
NOTE: I personally am against eugenics, because I am a Christian.
And I am in favour of making marriage and family a major focus of national policy; and I also believe that family size should be constrained by family resources - so, for these reasons - pro-natalism tempered by self-sufficiency, in general (with exceptions) I approve greater fertility/ larger families for the more wealthy than for the poorer. But (although in some ways similar) that is not eugenics, because it is not aiming to influence the gene pool.