A humidifier is made for this exact purpose.
I personally do not care for any form of a humidifier, and I tend to like my environment on the drier side (<30% humidity).
Only by using a humidity, and some common sense can you monitor the actual moisture content of the room. Most humidifiers belong to two groups, heat and ultrasonic. The first group simply heats water from a resevoir and allows the steam/vapor to enter the room. The second uses ultrasonic motion to disturb the water from the liquid into the atmosphere, without greatly affecting the temperature.
Most of these are meant to be temporary and mostly intended for those ill with respiratory infections or similar (unless they are part of a central heat/air system).
A better solution would be to identify the reason your room is cold, rather than run a space heater. These are notorious for causing fires, drying out the air, etc.
Any time the humidity starts to creep above 50% and/or a mist/vapor is deposited/concentrated there is a chance for mold/mildew.
Electronics generally do not like any form of moisture, but again, so long as you keep the humidity under 50% and prevent any misting or precipitation, there should not be any issues.
Prior to getting a humidifier or putting out a bowl of water anywhere, I would purchase an inexpensive humidistat to try and figure out what the moisture level is. Watch it for a day or so, then decide how to proceed.
Also, please be safe with your space heater. Many a desk, office and home have been lost to these things.