Older posts on related topics that might interest you:
In popular culture and meme posts there's a stereotype correlating living in Southern states and incest. Is there any historical reason why this stereotype is a thing? Why aren't Northern American states associated with this stereotype instead? by the inimitable /u/sunagainstgold
What are the historical origins of incest as a taboo? Was incest always taboo? If not, how did it become taboo? What were the historical arguments for / against the practice of incest?, where /u/Brigantus takes a very anthropological approach to the general issue of incest may also pique your interest.
There have been posts with genetic studies estimating various rates of cousin marriage across various regions of the country (perhaps here, perhaps on another subreddit), but I can’t seem to find them. I do remember that the rates decline over time. This article quotes an academic study of West Virginia that finds they declined significantly after the Civil War, and further after 1950.
The timing close to the civil war seems not be coincidental, as that’s the same period when when many states outlawed first cousin marriage. States that outlaw the practice generally chose to do so in the period 1858-1919. See this map here.
The practice seems to start on the frontier and to this day many Eastern states (both in the North and South) do not explicitly ban cousin marriages. Unfortunately, that particular article does not present any clear reason that this legal movement happened, though it was the start of a period with more regulation of personal status (for instance, anti-miscegenation laws banning interracial relationships) and more regulation of morality (for instance, the start of the temporance movement):
Perhaps surprisingly, these bans are not attributable to the rise of eugenics. Popular assumptions about hereditary risk and an associated need to control reproduction were widespread before the emergence of an organized eugenics movement around the turn of the 20th century. Indeed, most prominent American eugenists were, at best, lukewarm about the laws, which they thought both indiscriminate in their effects and difficult to enforce [2]. In the view of many eugenists, sterilization of the unfit would be a far more effective means of improving the race.
Nonetheless, in both the US and Europe, the frequency of first-cousin marriage—a practice that had often been favored, especially by elites—sharply declined during the second half of the 19th century [3]. (The reasons are both complex and contested, but likely include improved transportation and communication, which increased the range of marriage partners; a decline in family size, which limited the number of marriageable cousins; and greater female mobility and autonomy [4,5].) The fact that no European country barred cousins from marrying, while many US states did and still do, has often been interpreted as proof of a special American animosity toward the practice [6]. But this explanation ignores a number of factors, including the ease with which a handful of highly motivated activists—or even one individual—can be effective in the decentralized American system, especially when feelings do not run high on the other side of an issue. The recent Texas experience, where a state representative quietly tacked an amendment barring first-cousin marriage onto a child protection bill, is a case in point.
The laws must also be viewed in the context of a new, post–Civil War acceptance of the need for state oversight of education, commerce, and health and safety, including marriage and the family. Beginning in the 1860s, many states passed anti-miscegenation laws, increased the statutory age of marriage, and adopted or expanded medical and mental-capacity restrictions in marriage law [7]. Thus, laws prohibiting cousin marriage were but one aspect of a more general trend to broaden state authority in areas previously considered private. And unlike the situation in Britain and much of Europe, cousin marriage in the US was associated not with the aristocracy and upper middle class but with much easier targets: immigrants and the rural poor. In any case, by the late nineteenth century, in Europe as well as the US, marrying one's cousin had come to be viewed as reckless, and today, despite its continued popularity in many societies and among European elites historically, the practice is highly stigmatized in the West (and parts of Asia—the People's Republic of China, Taiwan, and both North and South Korea also prohibit cousin marriage) [8–11].
One thing to note is that it is taboo in much of the world (though by no means all of the world), especially peer countries in the OECD. Several American states took special legislative initiative against it, putting these taboos (some of them new) into law. The authors above suggest that this part of it may be in part tied to America’s decentralized political system as much as it is tied to America’s changing public morality.