If you have a "long neck" banjo I assume that you are tuned lower than "standard" open G tuning.
Start by capoing up and tuning to standard open G tuning [gDGBD]
Your "G" chord is then all strings open
Your "Em" is first and 4th strings fretted at fret 2
and for this song I would suggest that the best sounding "D" chord would be to leave strings 1 and 4 open and fret string 2 on fret 3 (it doubles up the D note of string 1) and fret string 3 at fret 2 (gives you the A from the chord). The "D" chord is of course incomplete, because it is missing the F#, but the tone of the chord fits in.

By the way, If I was accompanying said song I would only use the D and the Em chords.
Quack!!
Geoff the Duck.