Where to Stay
That really depends on three factors: how much time you've got, what you plan to do, and last but not least - your tastes and inclinations.
The area of Barrio Gotico and Las Ramblas is preferred by tourists, but I don't know anyone living in Barcelona who would want to live on Las Ramblas. Lively during the day and the center of nightlife, this neighborhood attracts tourists and pickpockets alike, so you always have to watch your belongings. In this sense, the area is dangerous, but it's highly unlikely that you'll get physically assulted here, or anywhere else in Barcelona. Having said that, I must warn you that the old town is more dirty and noisy than it is dangerous.
If you are here for just a few days and want to see as much as you can, then you should consider staying in Barrio Gotico. It would save you a lot of time on transportation. You can catch up on sleep when you get back home! :)
Barceloneta is another authentic picturesque neighborhood not far from the old town, but almost as dirty. It is close to the beach and known for seafood. If you are just two people travelling, you are most likely to find accommodation in this area because apartments here are smaller. Don't forget that Bcn is not such a big city (its fame is much bigger than its size), so what is important, in my opinion, is that the area is centric and well communicated by bus and metro to other parts of the city. Eixample, Gracia, and Vila Olimpica are such areas.
Eixample is a wider center of the city, and some parts of it are nicer than others. I like the areas of Paseo de Gracia, Sagrada Familia, and Parc de la Ciutadella.
Gracia is a popular area located above Eixample, and populated mainly with younger generation. It's relaxed atmosphere can be enjoyed on one of its many plazas with bars, restaurants and shops. Artsy and original, this neighborhood is lively but not wild.
Vila Olimpica is a posh area, located by the beach. With well-trimmed parks and recreational facilities, swimming pools, tennis courts, hiking paths, large shopping center and cinema in English, it is close to the Zoo and Barcelona Aquarium, and thus ideal for family holidays. It is an elite neighborhood with night clubs and casinos, also recommended to those who seek glamour, because that's as fancy as Barcelona gets!