I'm still not sure what was causing the problem, but it seems this particular monitor and desktop just don't get along. So I've moved some hardware around in my house and the problem is solved that way. May not be the best solution but it seemed easiest.
I have Googled this and seen many answers, but have not found one that solves the problem. Basically, my brother and I cobbled together the best parts from some old computers and I'm using the resulting computer to for specific tasks. The specs are (bad):
Windows XP SP3
AMD Sempron Procecessor
1.81 GHz, 896 MB of RAM
Radeon 9200 series video card
The motherboard is from an emachine but I don't know anything more specific.
It is scrolling if I use the original motherboard VGA port or the "new" Radeon one.
It is scrolling even in the BIOS.
Changing color, resolution, acceleration, etc. does not help.
Drivers are up to date.
It is not the monitor or cable (I've checked by plugging in another computer and there is no scrolling).
Any other thoughts?