Watch the Arpeggio, Basic online guitar lesson by Muriel Anderson from 50 Right-Hand Techniques You Must Know
Take some easy patterns to practice allowing the fingers to relax between strokes. These arpeggios are some of the patterns most often used to accompany songs.
It's important to watch your right hand to see that the fingers fall back to place immediately after each stroke.
Then, once the fingers get used to relaxing, they can go on automatic pilot and you can sing without thinking about what your right hand is doing. Usually the first bass note played is the root, the same note as the name of the chord.
Then alternate back and forth between another bass string, using your ear as a guide. Depending on the song and the speed of the arpeggio you can even do a double alternating bass with the thumb, for example strings 6, 4, 5, 4, or 5, 4, 6, 4.