The Airport of Barcelona is about 13 km from Barcelona center.
The best option for traveling to the airport from the cruise terminal is to catch a taxi. Taxis are quick and not expensive. The journey will take about 20 to 35 min's by taxi. The journey from the airport to the port is unlikely to cost you more than €40.
Alternative is a shared airport transfer from 11.25 EUR per person More information and booking
City Shuttle Bus
There is a port shuttle bus that goes from the cruise terminals to the Columbus Monument at the bottom of Las Ramblas. The bus is called Cruiser T3'. The fare is €3 per person. The nearest metro station is Drassanes which is on line L3. It is about 2-3 minutes walk from the Columbus Monument, where the shuttle bus stops. The bus is wheelchair friendly.
Taxi's are available at the pier - to city center will be around 15 Euro.