Answer:
Yes the color changes. If the star is moving away from the earth the color of the star is shifted towards red (red-shift).
This is similar to the change in sound from an approaching train whistle (higher pitch) or a receding train whistle (lower pitch) - the Doppler Effect.
Note that the equations for the Doppler Effect for sound and for light are different - sound travels thru air but no such medium can be identified for light.