Unless you are really pressed for time, or are carrying a massive amount of luggage, are physicially disabled or are traveling with a small child, there is no reason not to use public transportation to get from the airport into downtown Barcelona. You will have the option of either taking the bus, or getting on a free shuttle bus which will take you to the rail station (about 3-5 minutes away). The train costs €1.25 (but I was given a free pass by a woman in an official uniform the last time I took the train.) The taxi will cost between €25-40 (depending on traffic, time of day, and the driver's ethical outlook.) Once in downtown Barcelona (Pza. Cataluña or Pza. Sants) you might do well to catch a taxi to the hotel, just to avoid having to walk around a strange city with a suitcase.
I don't know about B&B's, but my folks have stayed at Hostal Subirats (Rda. Sant Pere, 62 Portal 1 Tlf. +34 933 191 881) several times and have liked it. (Best hamburger in Barcelona in the bar below.