We were in Naples both this past May and the preceding May. We did not have
major safety problems, and the police are very careful about warning
tourists. I am not sure we would return as there are parts of Italy, to say
nothing of Europe in general, that we enjoy more or have not yet seen.
However, I am delighted to have gone there. In general, my reaction is
similar to Grace Cattarossi's positive remarks.
We stayed at the Hotel Paradiso this past time, which I would highly recommend. It is a Best Western in the Mergellina district, convenient to the hovercrafts to Ischia, Capri, and Sorrento. Moreover, its view of the bay is superb, and it was not expensive (roughly $130/night for my wife, adult daughter, and self). You take a funicolare to reach it (or a cab). We stayed in a different Best Western the first year and would not recommend it either for location or service (we had to as we were traveling with a dog, which not every hotel would take).
Ann Widdowson suggested Sorrento as an alternative. It certainly is pretty and gives faster access to the Amalfi Coast and Pompeii. However, I personally like the atmosphere of Naples, particularly when I can eat a leisurely dinner at Brandi's. We wound up eating nearly every dinner there (our favorite waiter is Mario Scorzo--if you go, please give him our best). We were disappointed at other places we went to. Sorrento is far more geared to North Americans than is Naples itself, but it is also far more homogeneous.
I have a somewhat different caveat than most about Naples. If you arrive by train, you are pestered by dozens of people who spot you as a tourist. They commandeer all of the carts and actually make it difficult for you to get through. This is perhaps an understandable reflection of an area that is far more third world than the cities in the North of Italy. You also notice the poverty coming in by train, and there is far more begging. Do watch your belonging carefully.
Ira H. Bernstein, Ph. D.