Is there public transportation to the marina from Logan, like bus or subway?

There is also a water taxi from the airport that goes to the marina.

The T and Commuter Rail are within an 8-10 minute walk: North Station and Community College; bus line is 2 minutes away

The T can only get you so far, I'd probably recommend the blue line to State where you can pickup the 93 bus toward Sullivan cans get off at Warren St just a few hundred feet from the marina.

Subway