Sleeping Computer

Honestly that old of a system [that was pretty low end at the time it was sold] is not worth putting any money into fixing. If this were a high end workstation, then perhaps it would be worth investing some time/money.
You can do some basic troubleshooting [and at least it will not cost anything]
Do the following;
1 System OFF and pw cord removed from the pw supply
2 Open the side cover and pull the pw connections from ALL drives, both hd and optical
3 Reseat ie remove and replace both atx and aux/cpu pw connectors on the motherboard. While you have these OFF, give them a close inspection with a bright light. Look for any burn marks, arcing, etc. If they look dirty, you can give them a shot of contact cleaner. Replace both connectors and be sure they are fully seated
4 Remove the ram and blow out the slot or slots with a can of air. Reinstall the ram and again be sure it is fully seated
5 Clear cmos with the clear jumper. Read your manual for detailed instructions on how to perform this task
6 Replace the pw cord and attempt to pw ON. If the motherboard completes POST, then shutdown and start connecting devices on at a time until you find the failed part. If the board will not POST, it is really not worth putting any money into parts replacement.

Thank you for the suggestions. The new and old supplies both have good clean connectors. The old supply did not supply power on the 5 volt rails only. New supply checks out good on all voltages connected to motherboard. But I've never removed the three memory boards. I will do that now and see what happens. It didn't occur to me that bad memory or memory connection could prevent monitor and keyboard activity. I'll report back soon on the memory check. Hard drive is good--I removed it and installed Win 7 on it successfully on another computer before I submitted this question to you. Before I did that it would not boot up on the other computer. Oh yes, there was a little light brown colored dust concentrated in a small area right next to one of the memory boards which I vacuumed away. That should have tipped me off as a possibility of a problem there.