Just "boarders" doesn't work for me, as I immediately think of pupils at boarding schools or actors on tour in seedy theatrical digs. And children in the workhouse were also "boarders" -- they were boarded in rather than out (see 3rd link above).
A clear distinction between the "boarding out" of children and "fostering" doesn't work for the UK at least, as "boarding out" was the forerunner of "fostering", with the idea of providing as near to a normal home as possible for the child.
For example, the 1959 Scottish "Boarding-Out of Children" refers to these children as "boarded-out children" rather than "boarders" and specifies that it should be more than just providing accommodation and meals..
This act has, of course, has been superseded by more recent acts (Boarding Out and Fostering of Children, 1985 etc.) with the terminology becoming "fostering", and the "boarded-out child" becoming a "foster child" as dude suggested back in #1.
If this case is from the 19th century or even up to the 1980s, you could say that the child "was boarded out with a farmer's family".