Is it possible for my basement to be "too dry"?

I got a dehumidifier for my basement about a week ago and have been running it 24/7, with the water draining via a garden hose into the drain in the basement floor, and last night I slept in the basement because of tornado warnings in this area. This morning when I woke up, I had the same kind of sinus headache I've had before during the winter when the air is very dry, and as soon as I went upstairs and opened the door to the main floor of the house, I immediately noticed the much more humid air on the main floor, although it really doesn't feel humid when I just stay on the main floor. So that got me to wondering, is it possible for my basement to be "too dry"? My house is 110 years old, with exposed wooden floor beams comprising the ceiling of the basement, and the structure of the house is in excellent shape, but I got to wondering if running the dehumidifier 24/7 in the basement while we've had a lot of stormy weather will have any sort of negative effect on the basement, or if I should just keep it running.