I found these options :
From the Genova Brignoble train station there is a special "Vola Bus" which arrives near the taxi area on the same side of the street as the station. (The city buses stop across the street). It runs once or twice an hour, and you get tickets on board or go to the Tobacco shop inside the station and pay 10 eu cash. It goes right to the terminal.
You can take a train to a nearby area, but it seems you have a 20-25 minute walk (1.1 km) to the actual terminal. Maybe 3 eu. If I was traveling light with a backpack I might do that however from google map it seemed that that walk may be on the side of a highway which did not look good.
While getting on the volabus I saw a sign which said there was also a "vola taxi" which was only 8 eu to airport. However, it seemed to say there was a 4 person minimum. It was all in Italian.