Several of our servers are running OpenSSH 3.1, and are being prepared for upgrading. If I wanted to upgrade the SSH server on these systems, is it simply a matter of downloading the latest tar.gz file from the openssh website and running through a source installation procedure?

Well it depends on the system that you are using. One of the dependencies for Openssh is Openssl, it also has some serious security vulnerabilities, I believe that the current version is 0.9.7(some letter that I do not remember). You will want to make sure that this is upgraded before you apply your ssh upgrade.

Depending on the platform that you are running you may be able to acquire a prepackerd solution, or failing that you can download the source for Openssh (and Openssl for that matter) and perform the upgrade by compiling form source.