I only wanted to add that the most likely reason the dehumidifier isn't working during Winter months is because the humidity in Winter is so low, especially at latitudes such and the New England and New Jersey area.
I am currently at my home in Wisconsin, right on Lake Michigan, and I am putting over 8-10 gallons of purified water into my air because the humidity in my home, a condo, is barely above 30% ... in fact it is 33% right now and we get static shocks from each other and sure it's fun ... for a while ... yeah, let's shock daddy awake for the hundred millionth time! But this Winter is blanketing the US in cooler dryer air masses than recent prior Winters. My eyes are so dry at 33% humidity, yet even with 8-10 gallons per day, we can't raise our humidity, and our furnace uses a lot of humidity in combustion, as well, that is vented out the furnace exhaust.
Your humidifier may have internal presets or controls to not run at certain low humidities (the fan may, but the comressor and condensation coils may not), or, it's compressor and condensation cooling coils may not be powerful enough to pull humidity out of the air through the condensation process. My best supposition, though, is that your basement humidity is too low for the dehumidifier to pull moisture out of the air, and that your furnace system may be contributing to that process by simple combustion and exhausting of what little humidity there is left.
Thank you for visiting JustAnswer.com's Health Information. I hope that all of your concerns have been addressed and that your experience here was positive and helpful. If you have a new or different question, please don't hesitate to come back or submit a new question. It has been a pleasure helping you. Your FEEDBACK is very important to me.
XXXXX XXXXX, MD