Music for a While is in the key of A minor. Listening to the sung melody, the music is consistently diatonic which allows it to be settling and relaxing; the use of chromaticism (which wasn't common in Baroque music anyway) would have had the opposite effect.
At a glance, certain notes in the melody might come across as being chromatic but they actually just belong to the melodic minor scale.
Harmonically in a minor key, only the 7th degree of the scale tends to be sharpened which allows a minor tonality to function (see scale below). Melodically, this produces an interval of an augmented 2nd between the 6th and 7th degrees of the scale. When you hear a harmonic minor scale this part stands out in quite a striking way.
In Music for a While the melody is often based on a melodic minor scale which keeps the 6th and 7th degrees of the scale a tone apart, avoiding the awkward interval. This makes phrases such as "all" (bar 9, 0'47) and "wond'ring" (bar 10, 0'57) smoother, contributing towards the soothing effect of the music.
Perfect cadences are used regularly and these confirm the prevailing key very clearly. They are very audible and visible on the score.
Modulations occur in the B section to bring about variety and contrast including:
E minor, bar 15, 1'25 (B-E).
G major, bar 18, 1'43 (D-G).
C major, bar 22, 2'04 (G-C).