Do you have Rick's Snapshot Krakow, Warsaw & Gdansk? I didn't get to Gdansk, but it was invaluable for the other two. It's especially useful for questions such as these.
In it, he says that the bus #210 connects the airport with downtown, the main train station and the Main Town (2.80 zloty, more if purchased on board, 2 per hour weekdays, 1 per hour weekends). There's also the Airportbus shuttle going directly to various points downtown, "though the schedule is sporadic and it must be booked in advance" at http://www.mpapoland.pl (9-12 zloty depending on destination, 11 per day). A taxi takes 25 minutes and costs about 50 zloty. Note my copy of the book, from which I quoted this information, was published in 2012, so some details and prices may have changed.