Both brands make stable processors. AMD processors are typically a better deal, while Intel tends to focus on the high-end enthusiast market where AMD doesn't have as much competition to offer. Both are great companies, which is good, because they're the only two you have to choose from.
Define "Stable"... because using the traditional definition of Stability I would have to state that, although each brand has its issues, neither is realistically more "stable" than the other.
Hardware/Motherboard wise, however, Intel based motherboards tend to be offered in special models which use the best components available (something not offered for AMD users). These boards also cost an exorbitant amount of money.
Boards like the eVGA Classified Series and several Asus Republic of Gamer models do contain better components for their Intel socket based boards than their AMD counterparts.
The difference, however, will not likely show up in any standardized tests and, as such, you are not prone to see any differences less you overclock the shiz out of your rigs.
And then there is the current P67/H67 SATA-II chipset problem/recall to consider. This could potentially affect stability of any system which uses those ports.
EDIT: I should add that several AMD chipset based 790x boards have issues running RAM over 1066MHz. It doesn't affect every board but I have experienced it on several Asus and Gigabyte boards (resulting in Blue screens when attempting to run the RAM above 1066MHz). These are DDR3 based boards but built on an old and arguably obsolete chip-set so you're not likely to run into this problem.
^I'm not saying that they directly compete with the big guys from Intel, but MSI makes some great 'military class' boards, which from what I've seen, are built exceptionally well for the price. I'm sure Asus has their boards like this, too.