+1 on purchasing a hygrometer for the room, or a couple to scatter through the house.
+1 on keeping the instrument out of direct sun. Direct sun through windows, and landing on an instrument, will cause wood to fade over time. It does not have to be hot outside, or summer, for this to happen.
+1 on location choices
You can purchase portable wick/evaporator humidifiers (not ultrasonic or "cool mist" humidifiers) built by Vornado, Holmes, etc. and put them in the room with the uke. Some have digital controls so you can set the humidity to 45% or 50% and the unit will turn on and off as needed. Some folks have whole house humidifiers built into their HVAC ducts (April Aire, etc.)
Ultrasonic humidifiers used to cause white dust to settle on surfaces. The white dust is basically disolved solids in the water. The ultrasonic process breaks the water up enough that the particles essentially fall out. You don't have this problem with wick/evaporator humidifiers.
If you were to focus on the humidity in a single room, then you'd put a unit in that room and shut the door to that room. If you don't shut the door the humidifier will be conditioning air in adjoining spaces and will go through water much more quickly. To condition the whole house you just purchase more units and leave interior doors open.
We've used portable units scattered through the house for years. In the summer we use dehumidifiers. Between the humidifiers and dehumidifiers we are able to keep the house between 45-50%.