In terms of stability, for over a year Jenkins has offered a Long-Term Support (LTS) version for people who want to be more assured about the stability and support of the software they're installing.
Every three months or so, a previous release is selected which has been deemed as working well by the community of Jenkins users. This version is then branched, any important fixes (which have been "battle-tested") are backported into this Jenkins version, and then this release gets extra testing by various people and companies. Once it's ready for release, this becomes the new LTS version.
As new high-priority fixes come along, these are backported to the LTS version.
Numerous large users of Jenkins stick to the LTS line of releases, and according to the public Jenkins usage statistics, several thousand deployments are using it.
This should mean the LTS version you are downloading is even more stable than a random version chosen from the usual weekly release line.
Beyond the statistics, the situation regarding Jenkins usage, community size, its level of development, rate of new features added, number of new plugins and mailing list activity in comparison to Hudson doesn't seem to have changed (i.e. Jenkins remains ever-further ahead).
Basically, most of the points made in this previous discussion still apply, though the initial corporate support of Hudson appears to have subsided a little.
