The Pilgrims were undone by two second half goals on the 3G surface at Skegness Sports Association.
The scoreline had been locked at 0-0 at half-time, but Basford prevailed courtesy of strikes in the 46th and 70th minutes.
Boston (4-4-2): Oliver Swaby (Seth Greeves 50); Callum Rodger, Jack Withers, Taylor Gray, Sarok Ali (Arkadiusz Michalczyk 60); Marshall Young, Jordan Tate, Peter Evison, Jack Brown; Jenson Bark, Jenson Ball (Harrison Holland 70).