This is (hopefully) a fairly comprehensive list of instructions on how to install Adobe AIR relatively easily on 64-bit Linux Mint, both the main Ubuntu-based editions and the Debian-based editions.
First, download the Adobe AIR deb file from here.
Then follow the instructions here on how to turn it into a deb file that will install on 64-bit (and 32-bit) systems. (The instructions are for Ubuntu 10.10 but they work fine in 11.04 and its derivatives as well, so don't mind that.)
Before installing the application (the last step on the second page I linked to), install ia32-libs from Synaptic Package Manager (though you could install it however you prefer, I just happen to prefer the graphical tools whenever possible), since it's required in order to make 32-bit applications run on 64-bit systems.
After the 32-bit libraries are installed, you can install Adobe AIR using the deb you created.
Note that this only works this easily and nicely in Ubuntu and its derivatives. In LMDE (Debain-based) there are a few extra things you'd need to do to make Adobe AIR and its apps work as nicely as in the main, Ubuntu-based editions.
If you want to install AIR in 64-bit LMDE, you'll also need to:
Install ia32-libs-gtk (it's a separate package from ia32-libs in Debian, while it's included in ia32-libs in Ubuntu).
Download the 32-bit versions of libgnome-keyring0 (here) and libasound-plugins (here). (These files are required in order for certain AIR apps to securely save account passwords, and for AIR, and any 32-bit applications using ALSA for audio output, to properly use PulseAudio to play sound simultaneously with any other applications.)
Open those two deb files with a rooted Archive Manager (easiest way I find is to launch 'gksu nautilus' from the GNOME run dialog, then navigate to where you downloaded the files to, right-click them and select "Open with Archive Manager"),
Navigate to /usr/lib in each deb archive and extract the contents of that folder from the archive to /usr/lib32 on your system.
These extra steps will ensure that AIR applications perform (nearly) identically to how they perform in the Ubuntu-based Mint editions.
I hope this how-to can help a lot of people who might like to install Adobe AIR in 64-bit Linux Mint.