The top line is soprano and the bottom line is bass. Is there supposed to be some sort of harmonic function used by Bach by the use of C#?
The key signature is Bb Major, though this passage could also either be F Major or G Minor.
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This isn't really a matter of Bach 'harmonising a chromatic'. The melody doubtless came first. He's harmonising it WITH a chromatic.
A sharpened chromatic note will generally have the function of a leading note, the third of a dominant type chord. In this case the C# seems to be the third of A7, the dominant of D minor (assuming no key signature).
Maybe the music continues in D minor for a bit, in which case we could call a modulation. I suspect however that it's a very fleeting tonicisation of D, not worthy of that label. The chromatic note is just being chromatic. Adding colour between a pair of diatonic chords rather than leading us somewhere new.
In this instance, it being Bach and the C# is in the bass, I would guess that this chord would be a first-inversion A7 leading into a D major or minor chord. I doubt that it is a C# dim because in Bach-style chorale writing that chord would normally be in 1st inversion to avoid undesirable parallel movement. If the chord is a fully-diminished 7th, the inversion doesn't matter but that chord really came into use later than Bach.
In any voicing, the leading tone is normally followed by the tonic by half-step. In a secondary dominance like this one, the C# is acting like a leading tone into the D, so these chromatic movements are usually by half-step. Bach could have chosen a different voicing, but no matter what voice in which the C# was placed, it would be followed by D since it is "tonicizing" that chord. In this instance, the chromatic movement makes for quite a nice bass line.