The myxolydian scale is G-A-B-c-d-e-f-g and its transpositions. It is mode-degenerate with the major or ionian scale unless the 7th of the scale be present.

But IF the 7th is present, it must be a whole tone below the tonic above.

So if you want to harmonize a myxolydian melody in a myxolydian manner, you can't use any chord with an F# (such as D). You may use G, C, Dm, F. The G7 is formally OK but if the melody is old, using G7 might introduce an anachronistic sound. Another possiblity for harmonizing the d, besides Dm and G, is the following guitar chord:

3-2-0-0-3-X

Which is a G-chord with the top string left out and the top string's finger moved to the next string down.

An example of a myxolydian melody is "Orientis Partibus", or it would be except that most modern editions raise the 7th and turn it into an ordinary major melody. (Peter Paul and Mary used this 12-th century melody, with the raised 7th, for one of their songs, the Christmas song with "I said the donkey all shaggy and brown". To retrofit it to myxolydian, lower the 7th.)

T.