The area around La Barceloneta does not have a particularly high concentration of cultural venues. Nevertheles, if you're interested in Barcelona's history as an important Mediterranean shipyard, the Museu Marítim offers a very nice selection of boats, models, maps and paintings, plus a tour of one of the old ships.
The Harlem Jazz Club, near Port Vell, is a small dive presenting all kinds of interesting music: jazz, flamenco, rock, african, latin, fusion, experimental... along with Game-B it's a great example of Barcelona's diverse music scene. For a totally different vibe, the Irish Winds pub in the Maremàgnum complex offers live Celtic folk music.
Where to experience Barceloneta Culture:
For complete Barcelona culture listings (address, telephone, metro, etc.), click on the title links of each section above.