Does this buzzing noise happen only in certain display modes?
Does the pitch of it vary according to what mode you are in?
When it starts buzzing does it continue buzzing constantly or does it stop after a few seconds?
If the buzzing is only in certain low res, low scan rate modes (640x480 60hz, for example) this may actually be nothing to worry about so long as the image is still correctly displayed. It could just be that the deflection frequencies are low enough for you to hear. (Put your ear to a Television set sometime, you should be able to hear the vertical deflection as a soft snapping sound)
If it peters out over time, it might simply be a "resonance" in a coil or back voltage in the CRT, from mode switching. Again this is probably nothing you need to worry about, so long as the displayed image is clear and stable.
If it is in all modes at varying pitches. Usually this is an indication that the "Flyback" transformer, which supplies high voltage to the picture tube is becoming weak.
If the pitch is constant, it could be the power supply transformer.
In any event, you probably should have the monitor checked by a qualified technician. A good quality monitor should last at least 6 or 8 years... and it's worth the cost of one or two repairs to keep a good one going.