Since the song talks about the "House of the Rising Sun" being "down in New Orleans," we can infer that the singer of this song is somewhere up from New Orleans—somewhere a little farther north. It makes sense, then, that this song seems to have been sung quite a bit in the Appalachian Mountain region. It's true that we can narrow down the influences on the Animals' version to renditions that most likely took form in the American South sometime in the late-19th and early-20th centuries.
Also, it's probably worth noting once again that there's no real House of the Rising Sun in New Orleans on record.
But maybe the geographical setting of this song isn't that important. Over the last century, the song has been sung by musicians from all over America, and it was made famous by a British band. It's possible—though not proven—that the song might be rooted in English ballads that pre-dated the European settlement of America. The song has deals with the timeless, universal themes of personal regret and warnings of danger. It's bounced around so much precisely because misfortune, and pain, can happen anywhere.
Maybe there is something particularly Southern about the lyrics, though. Warning people away from a place in New Orleans is warning them away from a city, and in the South, cities haven't historically been the highly praised of places. In the days of the early American republic, Thomas Jefferson's agrarian vision was nursed on an isolated hilltop in Virginia, his idealization of the yeoman farmer was matched by his fear of the wage-earning urban worker, and his celebration of rural values was matched by his condemnation of urban vices.
Southern fears of the city were fed by the state of actual cities in the South: There were only a few, and there were very few roads or railroads making them readily accessible. In other words, most early American Southerners never saw a city of any significant size in their lifetime. In the decades after the Civil War, however, the insular character of the South began to break down. New railroads carried people to Northern and Southern cities alike—to Philadelphia, New York, Charleston, and New Orleans. With the introduction of the affordable car in the 1920s, travel became even easier.
It's not surprising, then, to find a song like "House of the Rising Sun" or "Rising Sun Blues" circulating with such energy in the decades surrounding the turn of the century. The warning issued to young men and women of the dangers down the road may resonate with people (especially parents) everywhere, but it makes particular sense in the American South of yesteryear.