One of my motivations for installing Linux Mint Debian Edition was to see for myself the efforts Valve have been making over the last few months, porting most of their games and their Steam platform to Linux.
It turns out that LMDE was not the best choice for this as the version of Linux that Valve primarily supports is Ubuntu, and there are a few differences between Debian and Ubuntu distributions - even though Ubuntu is more or less based on Debian.
One of the consequences is that the version of Steam distributed by Valve doesn't actually install on LMDE. Fortunately nearly all of the work to resolve this has been done already over here http://www.sarplot.org/howto/45/Steam_on_64bit_Debian_Wheezy_70. If you don't have an ATI graphics card those instructions might be all you need. If you do have an ATI card (like me) then when you try to launch steam, you may see the following error:
Error: You are missing the following 32-bit libraries, and Steam may not run:
One final step is needed, to have steam launching correctly:
sudo apt-get install libgl1-fglrx-glx:i386