We have some old CRT screens (we don't use them anymore, but didn't throw them away yet) that smell really bad when you activate them - why do they do that? I remember that this effect started relatively suddenly.
The smell could be from the electrolyte of leaking capacitors. I recently solved that issue for some CRT front projectors manufactured in 1994. One day one projector would emit a mildew-like odor shortly after turn-on, and the smell would dissipate after about 5 minutes. During board swapping for another issue, I was able to isolate the source of the odor and replace the leaking capacitors.
Depending on the formula for the electrolyte, I assume the smell could be different for other capacitors. The leaky caps in the PJ were mostly small (e.g. 100uF 16V) Nichicon KME and SP.
The electrolyte is typically corrosive. So the electronics will eventually fail, either from a leaky cap or a corroded joint. Somekind of corrective action would be needed if you want to maintain the monitors.
Looking for leaking capacitors is not easy. The tops of the can were not bulging. The worst ones had a brownish stain on the PCB, almost like flux residue. You need to look for stains around each capacitor or discolored solder joints (on the component side of the PCB). An ESR meter did not reliably detect some leaky caps
There are solvents to clean boards, but they are typically considered toxic, e.g. 1,1,1-trichloroethane.
Of course, exercise caution when opening up high-voltage devices.