A steady, high-pitched whistle from a monitor or TV set is usually caused by a faulty high-voltage component called a "flyback transformer."
The job of a flyback transformer is to take the 120 volts coming out of the wall and step it up to the 30,000 volts or so that the picture tube needs to operate. Often, a whistling flyback transformer is getting ready to fail; this usually causes sufficient damage that repair costs are greater than replacement costs.
It is sometimes possible to end the whistle by disassembling the monitor and using something like red silicone sealent around the base of the flyback transformer.
WARNING *** WARNING *** WARNING
Any TV or monitor uses 30,000 volts or more. This charge can be stored within the picture tube for long periods of time. If a picture tube discharges through your body, the results can be lethal. Never disassemble a monitor or attempt this kind of repair if you are not familiar with the safety requirements of working around high voltage in general and CRTs in particular!