The Port is listed as Puerto de Barcelona. If you are going from the international airport, you will have no trouble getting a taxi just outside the baggage terminal. The fare, a month ago, was about 13 euros plus add on for baggage and tip was less than 18 euros. Depending on the location of your hotel, taxi fare would be from about 7 to 12 euros. The port is in fact in the city, and you can see the terminal from the foot of the Ramblas. If you are going to the cruise port from the airport you might want to buy some euros before you leave home, or check out the location of the ATM in the airport. It's handy to have some one euro coins for tips handy.
When I put our cruise information together for our recent cruise out of Barcelona I had typed the Moll Adassat terminal information on a card so that a taxi driver would be sure to know that I wanted to go to the cruise terminal. I have taken a taxi to the terminal twice and have had no problem with the taxi driver knowing where I wanted to go. I was on a Celebrity ship but also saw a HAL ship in the same port. If you leave from the airport you will see the cruise ship to your right soon after you reach the coast.