I have Linux Mint 32-bit installed on a remote server. I was wondering if I can do the upgrade to 64-bit. I have access to SSH.
Recent versions of Debian (wheezy, currently testing) and Ubuntu (natty = 11.04) have some limited multiarch support: you can have amd64 and i386 packages installed on the same system. I don't know if Mint follows suit. But even on a the current Ubuntu (oneiric = 11.10) you can't have executables from both at the same time, only libraries and development packages. A cross-upgrade would be difficult if not impossible; I do not recommend attempting one if you can't easily access the machine's console.
I recommend a different approach:
Even then step 4 is risky; if all you need is to run a few specific 64-bit programs, stop at step 3. Or even at step 1 if Mint's multiarch support is sufficient for your needs (you need a 64-bit kernel to run any 64-bit program).
This is hard to do even with the box in front of you. Upgrading from 32bit to 64bit is difficult because there isn't any transition state that can work. It is an all or nothing affair.
So you are talking about a re-install. You may still be able to do this remotely however, in parallel with your existing installation. Without more detail about the current setup, you could do something like:
This sequence will likely need to be adapted for your setup.