The cheapest way is to catch a cab from right outside the airport. €35 down to the port.
Definately taxi from airport was there last week paid 30 euros from airport to port and 35 from port back to airport only takes 15 minutes or so.
Just be certain that it is Barcelona Airport you are flying into as flights from Bournemouth, which is near you, and other airports go into Girona airport and this is also called a Barcelona Airport and a taxi from there will cost about €100 each way. regards Neil.