Traveling into US Virgin Islands from mainland US--does not require any special visa since the US Virgin Islands are American territory.
Traveling from US Virgin Islands into mainland U--requires proof of status in the US, such as passport, green card, passport with US visa or other documentation to prove status.
Traveling into US Virgin Islands from other countries --considered to be similar to first time entry into US, so valid visa or proof of status needed.
Traveling from US Virgin Islands into other countries - nothing special visa-wise.
Passengers leaving from the US Virgin Islands into mainland US and passengers entereing US Virgin Islands from foreign countries will be subject to customs checking.
Note that US citizens DO NOT NEED A PASSPORT. You can travel to any US possession without a passport. Please check the Dept. of State web site.