It's the moisture that is coming out of your basement slab that is most likely causing the mold problem. You might consider using a dehumidifier in your basement, if the moisture problem is still manageable with such equipment. You might also consider increasing the amount of air chages/ventilation from your HVAC if you can. Otherwise, nothing short of relocating your bike storage to the living levels of your house above the basement will work as adding a moisture barrier under your existing basement slab and walls, if there isn't enough, won't be a practical/affordable solution for most.
Attic always round like a good place as it is mostly dry and dark, but unless you insulate and ventilate it adequately, you will just end up educating all the rubber, plastic and leather parts of your bikes = dry rot.
Your garage is actually not as bad as your basement as it gets enough natural ventilation just from the leakage around most garage doors and vents typically built on to them that limit moisture build up. Just make sure you hang your bike above the floor and don't cover it with non-breathable materials like plastic tarps, as they will catch the moisture migrating up from your garage floor slab....
leather saddles will also, pretty much mold up anywhere you store it, if it is put away wet. So drying as much as you can, gently with paper towels stuffed under them and such will be a good idea when you come in from a wet ride.