Temperature and humidity fluctuations can both cause problems with old wooden furniture. How much you want to protect against those two depends on how invested you are in protecting the furniture, and how delicate it is. Obviously a well-worn oak trunk is not going to require the same level of care as a priceless mahogany dining table with inlay.
Basements tend to be have pretty stable temperatures due to the enormous thermal mass of the soil, but you still want to make sure it doesn't drop too low in the winter or climb too high.
If you're in the northeast (based on your profile) this is probably a greater concern. You will need to deal with excessively high humidity in the summer and excessively low humidity in the winter. If you're serious about protecting the furniture you will need both a humidifier and a dehumidifier to keep the humidity stable, ideally around 50% and not fluctuating too much. [Side note: you mention a dryer affecting the humidity... if your dryer doesn't vent outside that's something you should probably correct, regardless of your storage plans. Improperly-vented dryers are a fire hazard, and the extremely humid air they put out can encourage mold, even in an otherwise dry climate.]
Unless you have signs of pest problems elsewhere in the house I wouldn't stress too much about this. If you want to add a little prophylactic treatment you could spread some diatomaceous earth around (non-toxic and safe for children and pets), which will help control insects.