I haven't heard that there are any particular areas to avoid in terms of crime, but some are obviously more attractive than others. Personally, if I were buying in the Port, I would focus on the Llenaire and Gotmar areas as these are both attractive and feel 'lived in' out of season. There are also some lovely apartment blocks at the Illa d'Or end of the Pine Walk. Some of the larger apartment complexes do seem a bit desolate out of season - as Lorenzo says, it's really important, when buying in PP, to make sure you like it all year round and that the amenities will be sufficient for your needs, as quite a lot of things are closed out of season.
We have had a couple of break-ins over the winter, when gangs roam the island looking for unlocked gates/shutters etc, so make sure that the apartment is as secure as possible, with lockable windows, shutters etc. I would also fit a burglar alarm - that said, PP is as safe as most places in Europe.
Deanscroft makes a very good point about heating! If you don't have some form of heating you will find that your apartment becomes quite damp over the winter months. The Mallorcans take this as a matter of course and seem just to paint over the damp bits when Spring comes - but it can be annoying to turn up in April and find that all your bedding, towels etc are grey with mould. To combat this we leave a window open during the winter (with the shutters locked) so air can circulate - but obviously this has security implications.
Sorry if this all seems a bit negative - just trying to point out some of the issues to bear in mind. If you do decide to buy you won't regret it, I promise!