The Barcelona cruise terminal is not very easily accessible from the airport, but there are cheap and easy ways to get there if you are willing to be creative.
There is a shuttle that serves the cruise port called the Blue Bus, which runs from the Cruise port terminals to the Christopher Columbus monument (pictured) and back. There is a close Metro stop that is a very short walk from the Christopher Columbus monument, and the monument is easy to find because you just walk to the sidewalk and look around and you'll see it. The Metro stop you want is Drassanes (Green Line, L3). The Blue Bus ticket costs 2 Euros per person, or 3 Euros for a same day round trip ticket. The round trip ticket cannot be used on different days, so for your arrival and departure from your cruise you'll need 2 one way tickets. Also, keep in mind the tickets can only be purchased with cash on the bus.
Another good way to catch the Blue Bus is if you are take the Barcelona Tours Bus, which is a hop on/hop off sight seeing bus. One of the stops where you can hop off is the Christopher Columbus monument, where you can catch the Blue Bus.
From the airport you can get to the Metro by taking the Barcelona Airport Train, specifically the C2 train to Passeig de Gracia. The train is a short walk from most of the airport, but if you are in Terminal 1 you'll want to take a shuttle bus. Once you get to Passeig de Gracia you can change to the Metro.