No. You should not continue using version CentOS 5.3, especially if you're running public-facing services. Security exploits are the concern here.
If you can bring your current OS revision to CentOS 5.8, or the most recent point-release, that will be the best approach if you want to stay on that major version. That is not particularly difficult, and most items should remain compatible. That's the point of these enterprise operating systems; to provide a stable target for X number of years. You should be find going to the most recent CentOS 5.x...
There are also a number of reasons to go to the 6.x revision of CentOS, mainly involving improved performance, compatibility, newer packages and being in alignment with developments in Linux... However, it is a big jump and would require a rebuild or reprovisioning of your system. Now that the 6.x branch has stabilized, it's become my default for new implementations.
There's no clean upgrade path.
As for whether to stay on 5.x or move to 6.x, you don't want to get too far behind. I still have a handful of CentOS 4.x server in the field, and while they fan fine during their supported lifetime (ending in February 2012), I'm now without updates and still have to do a full rebuild or data migration to get them to the newer versions of CentOS.
So it's simply a question of whether to go through the effort now or later.