1) Take the commuter rail from South Station. From Logan, take the Silver Line Bus (~2-3 bucks? About a 10-15 minute trip) to South Station. From South Station, you can take the commuter rail to Worcester (just read the signs and make sure you get a ticket for the right one!); I believe that's about $8. The downside, is this method will probably take the longest amount of time. The commuter rail (depending on what time of day you get there and if it's running with less stops or if it's stopping at every stop) can take almost up to two hours. But if you have the time like I did, a nice opportunity to catch up on some reading 🙂
2) The Peter Pan Bus. I took this on the way back from Worcester to Logan, and I think it goes directly to Logan but I can't remember for sure. If not it'll take you to South Station. I think it was about 10-15 dollars, but faster (maybe an hour trip, barring traffic) and a little more comfortable. You can also take this from Logan to Worcester as well I believe.
Do note that both of these options stop in Union Station in Worcester. Depending on where you're staying, it might not be that close and you might have to take a taxi to your hotel. I stayed at the Courtyard Marriott and it was probably about a mile from Union Station and the campus. Walking distance if you choose, but a short taxi ride as well.