Get a harmonica holder. Springsteen often plays harmonica over his guitar, and for this, you need the right gear. Practice strumming guitar chords and using the harmonica as it rests in its frame.
Follow the rhythm. Springsteen's harmonica riffs flow along to the driving rock beats in his songs. Listen to the way the harmonica goes in "This Hard Land" and other classic ballads where the full band plays.
Play melodic harmonica as opposed to "concept" harmonica. Although Springsteen plays along to rhythm, he doesn't use the harmonica as a rhythm instrument. Instead, he coaxes a defined pattern of notes out of the harmonica, generally in a minor key, that provide a great melody line over a part of a song, as in "Mansion on the Hill" (Nebraska) and other slow ballads.
Repeat riffs. When Springsteen finds a harmonica riff that works, he uses it multiple times in a song. Listen to the harmonica work in "Atlantic City" (Nebraska). The harmonica work is memorable in its melodic simplicity.
Get a "folk" sound. Listen to the way older, less electric bands play the harmonica as a folk instrument. Springsteen definitely takes his harmonica cues from an older tradition, building on its use as a melodic instrument and using it to evoke feelings of older times, the range of the cowboys or pioneer days.