How do you disable Internet Explorer

You can stop using I.E. but this is linked to RDP and explorer.exe also. You really can not disable something that explorer.exe needs to work. You can remove it from the windows components but it will still be on your computer..

Do you want to disable Internet Explorer only for remote desktop connections or in general?
The easiest solution would be:
Click on the Start button
Click on set program access and defaults
Select the Custom pull down list
Deselect the Enable access to this program checkbox for Internet Explorer
Click OK.

Now, all shortcuts are being disabled and iexplore.exe in C:\Program Files\Internet Explorer is hidden, but this can be easily overrided by typing iexplore in Start -> Run.
To disable the access on iexplore.exe:

Start -> Run -> gpedit.msc -> User configuration -> Administrative Templates -> System -> Don't run listed applications -> iexplore.exe.
Reboot.

I tried the gpedit.msc but when I click on IE it starts up again.

If you just want to prevent users from browsing WEB you can deploy "bugus proxy" technique explained here:

http://techrepublic.com.com/5100-1009_11-5838360.html#
and here :
http://pcsupport.about.com/od/quicktips/qt/disableie.htm

good luck
nedvis

Disable DHCP which won't allow your connections to see the internet and assign static ip addresses. Since you have static Ip addresses your computers will still be visable to your network and your remote connections won't be affected

If I remember correctly, remote desktop programs provided by Microsoft REQUIRES Internet Explorer in order to function, as stated in the first comment.
Should you choose to not use IE and still want remote desktop functionality, third-party software is necessary.

This seems like some good information at this point no action to disable internet access is required from me.  I believe they will be using some sort of group policy and accont access to segregate the computers.