Transfer from Barcelona train station to the airport.

Barcelona Sants Tren → Aeropuerto de Barcelona T2 Rodalies

It's the cheapest.
A train every 30', it takes approx 30' to Terminal 1 (a bit less to Terminal 2).
Departure: Estació de Sants
Destination: Aeroport T1

I'm not saying it's a better option than the metro, but another option is to exit
Sants, walk ~4 blocks to Placa d'Espanya, and take the Aerobus to either
T1 or T2.

In either case, you want to know which terminal you are going to before you
get there, as they are not colocated.

Apparently Renfe has a train R2 that runs from the Sants station and stops 200 meters from the T2 terminal at BCN Prat airport. It is cheaper than the airport bus. Not sure how convenient the Metro is from the train station. Plan to take the R2 train next week and will post how it goes.

I had a hard time finding the above info on the Renfe website.

When we arrived in Barcelona we purchased a couple single zone T-casual card at one of the Metro stations. The card is good for 10 trips on the Metro, TMB buses, as well as the local “Rodales” trains run by Renfe. We were able to get around on the Metro and local buses during our stay. We were also able to use the pass on the R2 Nord train from the Barcelona Sants train station to the Barcelona Prat Airport. The T casual card cost a little more the 12 euro and was definitely useful during our 3 day stay. I think of all of options to getting to the airport, the R2 train was the most economical and handy. The metro station a block from our place had no elevator or escalator which would have been a pain with a large suitcase. The train station was 5-6 blocks from our apartment.

Thankfully our flight took off in the early morning, so the train station almost looked like a ghost town compared to its typical zoo like mass of humanity.