Naples. Yeah. It's a tough topic for me because I like it and dislike it (love and hate are too strong) it at the same time. It is definitely worth visiting, I in no way want to discourage that, but there are some challenges. Some travelers seem to have real trouble with that, some find it energizing. I don't know if you can tell which camp you will be in until you are there.
First - about me, because I think the source of advice like this is important. I travel a lot in southern Europe and like a lot of places some people consider a little gritty. For example, I love Sicily and did not find Palermo unpleasant in the least - I enjoyed every bit of my time there but I see where it could be difficult for some. Parts of Granada, Spain are considered questionable by some travelers but it's my absolute favorite place in Europe. I am a city person, I also love Rome and Athens (including areas outside the tourist core) and frankly don't quite understand why some people find them overwhelming. I don't like places that are too "polished" or made up for rich tourists. I thought going in that, given all that, I'd really like Naples. Just didn't quite happen for me.
I spent 4 nights in Naples recently, and also went years ago for maybe 2 days. I'm ignoring my older visit and focusing on what I saw recently.
Naples - the positive. It has some amazing sights that are not as overrun with tourists as similarly interesting sights in other parts of Italy. Most of the people are very friendly. There is a real vibrancy to the street life that can be exciting. The pizza is excellent. The city is a good jumping off point for places like Pompeii, Amalfi coast, Herculanem, and Mt. Vesuvius.
The negative - I don't know, it just wore me down in a way Rome or Athens doesn't. Very few trees in the historic core or many other neighborhoods (I don't mean parks, I mean even those little city trees planted in sidewalks or around squares). It wasn't as pleasant to wander as other cities. I felt more like I was going from sight to sight (all the sights were great!) and not enjoying the time between sights. That's not normal for me. I like to soak in the vibe of a place, walk around, take random detours and enjoy being out in the city. That wasn't pleasant for me in Naples and I just can't put my finger on why.
Safety - I think reports of safety concerns are a bit overblown if you use common sense. I never once felt physically unsafe. I felt worried about pickpockets on the Circumvesuviana train, there were some suspicious people roaming the cars, but it was fine. I think I got ripped off by one cab driver, but that's my fault and common anywhere.
I stayed in the Chiaia neighborhood, not far from Piazza Plebiscito. This was a nice area, a little removed from the hubub of the historic core but easily walkable to it, also walkable to the waterfront. It's considered a wealthier district of the city, but it wasn't expensive or overly fancy. I liked the area. I'd stay there or in the historic core area (toledo/dante metro stops vicinity).