Use OpenJDK because it do not contain the closed-source Oracle browser plugin that have been the attack vector for all the latest zero-day exploits.
If you want a JRE with included security updates install:
OpenJDK builds in Ubuntu are based on IcedTea, this is good because the IcedTea team apply the latest security updates, that fix issues inside the core java libraries, on top of OpenJDK 6 and 7. You can find all IcedTea security announcements in gnu_andrew's blog who maintains the IcedTea security releases.
If you want a secure java browser plugin and webstart then use the IcedTea-web plugin and javaws. You install IcedTea-web by installing:
IcedTea-web implements a more strict security policy compared to Oracles closed-source plugin:
By using the above mentioned IcedTea-OpenJDK packages you get all new security updates when performing regular Ubuntu updates. If you manually install Oracles closed source releases you have to update manually.