My boss (on W98) let the automatic update do its thing and
start the DirectX update today. It appeared to be
successful and prompted for a reboot. When he reboots
now, he gets the network login, then the desktop appears
for a split second and then is gone leaving his background
and a cursor. How can I recover from this for him?

I guess when you get to the blank screen with the curser
you do not have a Start/tool bar on the bottom. If you
did? You could select "Restart in MS-Dos" and do a
scanreg/restore. Try restarting computer and hold "Ctrl"
or "F8" key. You will get a selection screen. Hit "Esc"
key to stop the timer. Here you can start in safe mode. I
have not been there in eons so I can't remember if there
is a selection to start in MS-Dos. If there is? Select
that and do the scanreg/restore to a previous date before
the download.
Or... if you have a boot disk? Then plug it in and start
computer. It will give directions on what is needed to do.
Good Luck!