Since 9/11/01, a new "security" measure requires that everyone's proof of citizenship is checked when arriving on St. Thomas (actually, when arriving at any US port after having called at a non-US port). This takes place early morning on Wednesday, after the ship is tied up at St. Thomas. Each deck is given a time to report. It usually goes pretty smoothly, but every once in a while there are people who drag the process out.
Still, I've usually been able to get off the ship by 8:30. The Renaissance is over on the north east shore, near Coki Beach, so you don't have to go all the way through downtown to make get there.
Tendering happens every so often, depending on how many ships are in port that day. It is more likely to happen during the winter, when there are more ships in the Caribbean, but you probably have a 1 in 3 chance of having to tender. If so, you can grab a cab from the tender pier downtown.