1.3.9g had some irritating little bugs (one was in the HTML editor feature) so you should at the very least upgrade to 1.3.9h . There is no database upgrade needed from g to h ... only a few code files.
And - in general - when a new version is released, one should go about the procedures for an upgrade. Just what these "procedures" are depends to a large extent on the nature of the add-ons which are operating in your installation.
There are still several add-ons that are not yet compatible with 1.5, so if you use those not-yet-converted addons, you have a choice at this point:
- remove the incompatible add-ons and upgrade to 1.5
- wait for upgrades to your add-on (or re-code them yourself)
1.3.9h remains a stable release, with no known (or serious) vulnerabilities. 1.5 was built to introduce a range of strict security situations that are now required for PCI/DSS and PCA/DSS compliance. The nature of these measures has resulted in some significant alterations to the way the ADMIN area operates - and it is this main area where some add-ons fail.
You can (reasonably safely) maintain your 1.3.9h version for a while yet... but there ought to be a strategy in place for you to upgrade.