There is no easy way that I know of to turn  off this feature. The
only way I am aware of would be to edit the Upload.aspx file  for each
site. In that file you will find a JavaScript function called
_spBodyOnLoad() In that function you will find an iff statement that
looks like this.

if (new ActiveXObject("STSUpld.UploadCtl"))
{
hideLink = false;
}

change the hideLink value to true. This will hide the multiple  file
upload link even if the ActiveX control exists.

Of course users will still be able to upload  multiple files using the
Explorer view.

The Explorer view can be turned off.

we have yet to be able to find this
JavaScript code in any of our Upload.aspx files... Do you have any
other suggestions on where to look for this code or any other way we
could disable  or remove the multiple  uploads link? Our customer really
would like this feature  disabled.

Where are you looking for the upload.aspx file? Its located in the Layouts
directory under the Template directory in the 12 hive. There is only one
upload.aspx file. The function you need is in the first <Script> tag.
This is the only way that I know to remove the multiple  uploads link.

However, as I already pointed out you should let your client know that
removing this link will not prevent users from uploading multiple files
using DAV. You can't turn  off DAV support entirely because Office apps use
DAV to save files to a Document library. So using the Windows Explorer view
you will always be able to upload  multiple files. If you remove Windows
Explorer view from the menu a user can still open it by opening the document
library as a Web Folder.

The point of all this. You can't prevent users from uploading multiple
files. Your client would be better off training users not to upload
multiple files than preventing it.

We did eventually find the correct
Upload.aspx file  late yesterday. (There are also multiple  Upload.aspx
files in each Forms folder.)

We do understand that it is impossible to prevent users from uploading
multiple files if they are determined :) We wanted to at least remove
the most obvious link.

The upload.aspx files in each document library in the forms folder should be
ghosted copies of the one in the 12 hive. To make a global change edit the
one in the 12 hive.