If you're playing in G-mixolydian, you want to avoid the D-major chord, just as Sorcha noted. If you're playing in D-dorian, avoid the B-flat and any chord containing it.

When I play in G-mixolydian I try to use the F-major chord several times if it fits the air, especially at cadences, to emphasize the mixolydian quality.

More generally, you might find it a useful experiment to try harmonizing an air with only the notes that are in the air (and their octaves, obviously.) If the air is hexatonic, try playing countermelodies that contain only the same six notes. For pentatonic melodies this can be tricky. I once harmonized a pentatonic air in 3 parts, and found that the results were not satisfactory unless the lowest part confined itself mainly to elongated notes, while the shorter notes appeared mainly in the upper two parts.

T.