The line you are seeing is caused by a 60 Hz noise somewhere near the monitor. When the horizontal scan rate of the monitor is 60.15Hz, it is very susceptible to external noise in the 60Hz area. The only solution is to find the source of the noise and move it or move the monitor. The noise could be from a lamp or anything else plugged into a nearby outlet.
You may also want to try turning the monitor in different directions to see if things improve at all. The interference may change due to magnetic fields.