I know, for instance, that RNA is much more succetible to alkaline hydrolysis than DNA and this difference is determined by the presence of 2'-hydroxyl group in ribose. I have also heard that "DNA is more stable" and thus was selected as the repository of genetic information. Though, I still haven't understood why the lack of just one hydroxyl would lead to more stability...
As user137 said, the general base abstracts a proton from the 2'OH and subsequently the 2'O- renders a nucleophilic attack on the δ+ Phosphorous, leading to the hydrolysis of the phosphodiester bond. There can be slight variations in the mechanism and the intermediates; for details see this review.
After hydrolysis both 2'- and 3'- phosphates can form.
RNA is not thermodynamically unstable; apart from being unstable in the presence of a strong base RNA is biochemically unstable too. This is because there are several RNAses and most of them degrade RNA by a mechanism similar to alkaline hydrolysis (they cannot act on DNA). Also see this post.