Kaleboel links to this excellent webpage called "Street Scams in Barcelona". I've never seen it before, but it's been around for seven years and is full of first-person testimonials.
Petty street crime in Barcelona has always been bad, but it's gotten worse over about the last three or four years now. Violence, now rare, has become common. You will not be murdered--I have never heard of a tourist being murdered in Barcelona, a city with only 40-50 murders a year, almost all domestic violence. You WILL, however be a target of Barcelona street criminals while here. Somebody WILL try to take you off, no matter how cautious you think you're being or how travel-savvy you think you are.

Some British guy (his name is Terrence, so he ain't from, say, Indiana) on the Street Scams thread suggests that one should avoid dressing like an American tourist. Doesn't matter. They'll pick you out as a tourist anyway, no matter if you're wearing the Official Grayish-Black Barcelona Uniform or not. He also suggests that one should not talk to strangers in Barcelona, just as one would not talk to strangers in Atlanta or Chicago. Terrence has it just the wrong way around: Americans WOULD talk to strangers in Atlanta or Chicago, because anywhere travelers go--the downtown streets, the tourist attractions, the shopping areas, attractive restaurants and bars, airports--in the United States is safe. Bad things in America happen in places far away from where tourists or business travelers go. The problem is that bad things in Barcelona happen exactly where tourists go.

Most of the city is very safe from petty crime. But the tourist areas are definitely not. All of the Old City, inside the Rondas south of Pl. Catalunya between Paralelo and the Parque Ciudadela, is dangerous; so are the areas around the Parque Guell and the Sagrada Familia.

Remember that ninety-seven percent of Spaniards are wonderful folks, but the Spaniards who earn their livings dealing with moron-tourists are not. They deal with so many morons that they assume all foreign tourists are morons. Even if you're not, you will be treated like one. The locals will give you no sympathy if you get taken off on the Ramblas. If you somehow get in trouble away from the Old City, though, the people you'll deal with aren't used to dealing with moron-tourists and so will consider you to be a real person.

How to Avoid Being a Victim

1. Stay in a hotel and eat in restaurants outside the Old City. The Eixample, Sarria, St. Gervasi, and Gracia are all very nice and outside the high-crime areas. Sure it's more expensive. It's cheaper in the long run. And the food's better because real locals eat in those places up there.
2. When going touristing in the Old City, leave everything in the hotel safe. Bring a twenty-euro note and a disposable camera. No passport, no credit cards, no fancy equipment, just enough for a sandwich and a cup of coffee.
3. Watch your ass when arriving with your bags and leaving with them. That's when you are most vulnerable. By the way, travel light.
4. Avoid gypsies and Arabs. Assume they are thieves until proven otherwise. And don't stay around them long enough to give them a chance to prove anything. Damn right I'm prejudiced.
5. Avoid anyone who stops you and tries to get your attention, especially if you are in a car anywhere or on foot in the Old City. Let 'em get the time or a light or directions from someone else. Just shrug and move on. If the person becomes insistent, move on really fast.
6. Don't give any money to street performers, and don't stop and watch them. That's when your pocket gets picked.

Scams and Robbery Techniques They've Tried on Me

1. Mustard-on-your-shirt scam
2. Gypsy-women with carnations scam (dozens of times--just get away)
3. Asking-for-directions scam
4. Asking you the time scam
5. Asking for a cigarette scam
6. That American woman who begs near the Plaza del Pi
7. Asking for a few euros for a train ticket (maybe 200 times)
8. Trying to grab your card at an ATM
9. Offering to "give" you a cigarette lighter
10. Trying to pick your pocket in the subway (uncountable times)
11. Begging with doped dogs as alms bait (uncountable times)
12. The "football scam"--one kid shows you Ronaldo's moves while another robs you
13. Surrounding you and overpowering you (four or five times--never got me but two friends, yes. One guy whacked me with a metal bar while I was getting away.)
14. Taking you down with a stranglehold from behind (twice. It worked once.)

Two that I haven't seen mentioned yet:

1. They say they're from a school for the deaf and show you some phony ID. Then they sell you some fifty-cent piece of leather crap for five euros. I have seen this scam in both Madrid and Barcelona.

2. They set up a "petition drive" and ask people to sign against drugs or against AIDS or something else that everybody's against. While you are signing and your attention is distracted, your pocket will be picked.