As small point to the debate; to those who are old enough to remember, the Kingston Trio's first recording was a 45 with "Tom Dooley" on one side and brownie points to anyone who can name the other side... To my knowledge the Weavers never did.

Clearly, though some would try any credit the Weavers with being the most influential record album, as one who was in his teens during this period, I was aware of both at the same time and had both the albums, and as I traveled from Alexandria, Virginia to Cleveland, Ohio to Aiea, Hawaii during that year 1956-57 (my father was a Naval Officer) the Kingston Trio had far more influence. They had air-play all over he place and without payola. As noted above, they made it commercial, accessable, well-known and desirable, which may be crass, but also fits the best definition of "Influential".