HSL travels to the Helsinki Airport by train. Choose I or P both travel to the airport from the City Center, train stops at Airport so you cannot miss it. Helsinki Airport lies at the sector C so make sure you buy a ticket that reaches to your destination.
A taxi to or from the airport will cost upwards of €40 and probably won't even be that much faster. Don't confuse the 615 with the Finnair bus which takes about the same time and is a few euros more expensive. The Finnair bus leaves from the post office side of the station, the 615 from the other (Ateneum) side.
Check online if your flight is departing on time because Finnish aviators like to strike every once in a while.