How to get to El Born
The Yellow L4 Metro line takes you straight into El Born from anywhere in the city. Barceloneta and Jaume I metro stops are the ones you need. Alternatively use the red L1 metro line and get off at Arc de Triomf.
From the Airport
If travelling from the airport take a taxi, that is the most reasonable way to reach El Born. There is a queue outside the airport, you cannot just hail a cab from the road. The minimum price for a taxi is 20 Euros, but there are also tolls and supplements to pay for luggage.
The big advantage is that it will take only about 20 minutes to arrive anywhere in El Born.
There are two bus options from the airpot to the city. "TMB Airport day bus (No.46).", this is a red bus that stops at both terminals of the aiport. Find more details of it here: http://www.tmb.cat/en/linia-de-bus/-/bus/46. It won't take you all the way to El Born though, you need to travel 14 stops to Plaza Espanya then change to the Metro for the last part of the trip.
Secondly there is also an Aeroporto bus, but this has the same problem in that you will need to change.
The major train station Estació de França (French station) is just outside the district at the east and is within walking distance of all the interesting parts.