Strictly speaking El Born ends once you cross C/Princesa, but the geographical limits of a trendy neighbourhood tend to expand exponentially ! These days, La Ribera running up to Santa Caterina market is probably cooler and a bit edgier, but Passeig del Born and the area around Sta Maria del Mar is lovely to walk around & great to eat and drink.... if a little pricy by local standards.
Personally, I'd head to Plaza Allada Vermell , just across C/Princesa for a cheaper beer & a more local less touristy atmosphere. El Born proper is more style orientated & has nice shops but fashion is fickle and people move on..... stay north of Princesa .... up to Palau de la Musica Catalana if you want a bit more edge ..... but as ever in Barcelona , watch your wallet, gold chain, bag, mobile phone etc, etc..... thieves abound & they're really very good at what they do..... unlike the Spanish police.
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