There are many local trains from Helsinki and the whole region going to or through Vantaa. It is the fastest easy connection and journey last about 10 to 20 minutes depends on the locations.
Vantaa is a large city, if you take a taxi from east to west, you can easily get to 40-50 € a ride, so, if possible, use a train or bus.
The Helsinki airport is actually in Vantaa and it called Helsinki-Vantaa Airport. It is rather small but very advance in technology. There are no long queues and arriving & departing is very fast,