My friend plays an online computer game called World of Warcraft. Due
to a high level of 3rd party programs people can use to 'cheat' they
have developed some anti-cheating software.

What this does is scan ALL your running programs, inlcuding email,
open web browsers etc. Full information is found on this foreum (you
need to register but it is free)
http://www.wowsharp.net/forums/viewtopic.php?t=7024

Is this legal ? Say for example I am logged into online banking they
now have knowledge of my bank information. So now they have full
access to any information you have stored on your computer and can
access it at anytime without your knowledge.

thanks

Wow666

When installing the software your friend accepted the EULA (end user
license agreement).  The EULA would need to be examined in detail to
determine if the company is invading the users privacy, or if the user
gave up certain rights to privacy by accepting the EULA.  To learn
more about EULAs visit the following web site:

http://en.wikipedia.org/wiki/EULA

Another thing to note, the Blizzard software, called Warden, does NOT
monitor program activity.  It scans your active processes (running
programs) and creates a hash, which is then checked against known
cheat programs.  If no cheat programs are found, Warden does not
communicate back to Blizzard.

So not only can they not view what your programs are doing, but they
can also not even see WHAT programs are open.  (I read your link, and
it is accurate, Warden does access this information.  However, as I
stated, all information is first hashed, compared, and only then, if a
cheat is detected, is that info sent to Blizzard).