To harmonize a melody, the musician simply plays a chord on the first beat of each bar that contains the melody note that occurs on that beat. In * time it is also possible to add another chord on the third beat of the bar.
The Banks of the Ohio is harmonized with the chords C, F and G. This is a good starting point for finding vocal harmonies for the song. If you play guitar or keyboard, you could experiment using notes from these chords until you find a harmony that suits your voice.