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You DO need to be careful - Naples certainly has something of a crime problem and you need to be on your guard with your valuables, don't flash expensive jewellery around, etc.
As a tourist you're going to be a slightly obvious target thieves know you're likely to have a camera, for example. As for the prostitutes and drug addicts, etc - well, all cities have those! It will be obvious if you find yourself about to enter a REAL no-go area. Use your instincts and don't go down quiet dark streets. The biggest problem in the centre is pickpocketing and bag snatching really.
Just make sure that your bag is in front of you preferably across your body and zipped up, don't keep things in back pockets, etc. Be wary of people who "accidentally" bump into you and of groups of children begging.
And you probably want to walk on the part of the pavement furthest from the road to avoid the thieves on mopeds who'll grab your bag or even cut the straps to do so. BUT, having said all that, it really isn't that bad as long as you're careful. I don't know who told you not to use taxis, but really that shouldn't be a problem. I know a few Neapolitan taxi drivers might try to rip you off, but remember that they're supposed to have a list of fixed fares in their cab and if they don't show that to you and offer you that fare, don't get in.
Obviously there's still the possibility that when you get to your destination they'll try to charge you more, but be firm helps if you speak Italian. And on the buses and the metro, again, it's just a question of keeping a close eye on your possessions and being sensible. But you might not come across anything like that. If you look back through the Naples forum there's a trip report from someone who went recently who said they didn't have any of the problems that they'd been warned about Naples is an amazing city.