How prevalent eugenics was to your life in America depended on where you lived and who you were.

For example, eugenic ideology played a role (the size of which is disputed) in the passing of the Immigration Restriction Act of 1924, which limited immigration from Eastern and Southern Europe. Trying to escape the Nazis, and from one of those countries? That's going to make escaping to the USA (in particular) hard for you. (This comes up with regards to Jewish emigration.)

Are you in a state institution (e.g. for the mentally ill or deficient) in state that has active and vigorous sterilization laws? You might run the risk of getting sterilized (vasectomy for men, tubes tied for women). Some states were heavy sterilizers (California was the heaviest, though it was on the decline in the 1940s, but continued to sterilize at high rates until around 1949; Virginia, North Carolina, Georgia, and a few other states were starting to sterilize a lot in the period and would continue to do so through the 1950s and early 1960s; some states had no sterilization laws at all or didn't use them at that point).

If you lived in certain states, and wanted to marry someone of a different race, there were sometimes eugenic aspects to the laws prohibiting you from doing so, as well.

These are all of what are known as the "negative" eugenics policies, which is to say they were aimed at prohibiting certain types of procreation (not that they were "bad"). There were also "positive" eugenic policies, meant to encourage certain types of procreation (e.g. among the "fit") though by the 1940s a lot of these had gone out of style. This is a much broader and sometimes vaguer category. For example, one of the founders of marriage counseling in the United States was a major eugenicist — Paul Popenoe was a major advocate for sterilization of the "unfit" in the early 20th century and became a major advocate for salvaging the marriages of the "fit" in the later 20th century. Is marriage counseling eugenic? Well, you can do it for eugenic reasons, but it's not clear that we want to call the whole institution "eugenic."

But the basic point is that in the USA, the prevalence of eugenics was actually pretty limited to a few specific locations and types of interactions. It was not widespread, even its heyday. If you were in one of those locations, it could impact you in major, permanent ways (e.g. sterilization). But if you were not, you might not even know it was going on.

On the whole, these kinds of "hereditarian" attitudes were definitely on the wane in the United States from the Great Depression onward. Not vanished, and not eliminated, but the economic disruptions of the 1920s made it clear to many people that things like crime, poverty, and disease were related to far more complicated factors than Mendelian genetics. Biologists as a community had already turned away from some of the more extreme forms of eugenics, like sterilization, because it was clear that the genetics were too complicated to be affected much that way (see e.g. the Hardy-Weinberg equilibrium, which was a major objection to the idea that sterilization of people in institutions would have much affect on the total gene pool expression of anything).

There is a question of how much we want to link racism in general to eugenics and vice versa. Racism was certainly prevalent, and in some places got much worse in the run-up to WWII on account of the Great Migration. Race-mixing was seen as abhorrent by many. Did they put a scientific veneer on this? Sometimes yes, sometimes no. Do we want to give the ideology of eugenics credit for all of this? I'm not sure. Eugenic language did sometimes come up in these kinds of discussions (e.g. Lothrop Stoddard) but I would personally put interwar racism in a somewhat different category.

OK, so how about the change after the Nazis? The Nazis did present anti-racists with a great opportunity to disprove that racism had a scientific backing. They eagerly lept upon the ridiculous Aryan supremacy theory and used it as one of the charges of Hitler being a madman. It proved a great opportunity for anthropologists of the Boas school to produce a lot of work both before and after the war about the evils of racism, the lack of scientific basis for the term "race," and things like that. They were also able to mobilize the UNESCO organization for this work after the war (and not uncontroversially, amongst scientists). Look up Ruth Benedith and Ashley Montagu if you want more on this.

The discovery of the death camps certainly helps discredit theories of racial superiority, coined the term "genocide," and in the USA certainly made anti-Semitism pretty taboo. Did it lead to an immediate distancing with eugenics? No. The connection between what the Nazis did and eugenics was not a firm one, cuturally, until the 1960s and 1970s. In the 1940s they saw what the Nazis did as racism with a scientific veneer, and there were those who believed that eugenics could be still a real science despite this (in the same way that the Nazi embrace of anti-smoking campaigns did not necessarily mean all anti-smoking campaigns were Nazi). I have seen no evidence that Nazism had any influence on US sterilization rates, for example — the war did have a disruptive effect because many mental institutions got used for war purposes or otherwise had population disruptions, but states that were already declining in sterilizations before the war just continued on their paths, states that were beginning to embrace sterilization continued for many decades. In California, sterilizations really ended around 1951 and not a single person involved in that — including the newspapers — ever thought to make a connection between what they were doing and what the Nazis did. Those connections came later.

Some of the professional societies relating to eugenics did start to shift their focus, though. "Genetic counseling" spun out of eugenics altogether around this period, and they deliberately chose tenets that they thought would distance them from eugenics, which had acquired the taint of quackery. The Journal of Eugenics renamed itself the Journal of Heredity around this time. How much of this is because of the Nazis, and how much because of other trends? Hard to sort out.

So the short answer is that the perceptions were already changing in some ways well before the Nazis, and by the time people associated eugenics with Nazism, the perceptions were already mostly changed. Which is not the standard, neat narrative you will normally get, where people learn from the sins of the Nazis and change their minds overnight.

The best short book on this is Diane Paul's Controlling Human Heredity, which is one of the few books that actually tries to document when these attitudes changed (many other books just assert it was post-WWII and take it for granted how it happened). On California's sterilization story, I will do an obnoxious thing and self-cite: "States of Eugenics: Institutions and Practices of Compulsory Sterilization in California". The article (if I do say so) makes an argument about how may histories of American eugenics fundamentally fail to take into account the nature of how the gap between ideology and practice was mediated by institutions, and what implications that has for the US context, especially the differences between the US and Germany.