AMD CPUs are every bit as stable as Intel CPUs, and (at stock speed) perform generally much better than Intel CPUs available at the same price. Everyone who claims the opposite is either ignorant or malicious.
Having said that, the stability of a PC is not only a function of CPU stability; a very important role is also played by the motherboard and, in particular, the chipset. Intel has been producing very good, very stable chipsets for a long time. This has contributed greatly to the good reputation for stability that Intel has been enjoying.
By contrast, most AMD systems have for some time been based on VIA chipsets. These aren't as good as Intel chipsets; some of these were really quite bad, like the VIA 266 chipset. 266A and 333 are much better, though probably still not as stable as Intel chipsets.
The bad reputation for stability that AMD has suffered from is mostly due to unsatisfactory chipsets made by VIA. The chipsets that VIA makes today (266A and 333) are actually not bad; apart from this, there have been and are alternatives, for instance chipsets made by SiS (735 and 745) and nVidia (nForce). These are extremely good, every bit as good as Intel chipsets, and less expensive.
If stability is a mjor concern for you, by all means go for an AMD CPU, but you might want to avoid VIA chipsets, in favor of SiS 735/745 or nForce. This gives you a reasonably broad choice of good, inexpensive motherboards, for example Asus A7S333, MSI SiS 745 Ultra, or Abit NV7-133R or MSI K7N415 PRO.
In my experience and the experience of many others, an AMD based system with any one of these motherboards (or for that matter one with a VIA 266A or 333 based motherboard) will more likely than not be rock solid, will perform very impressively, at an astonishingly low price. Go for it.