Congrats on the upcoming new addition to your family. My 6-week old daughter is in her swing right in front of me right now while I'm typing this. Absolutely amazing! Nothing like parenthood.
Your best bet is to store them properly in a climate controlled condition. If you don't have enough space for them in your house, I would look into the cost of a nearby climate controlled storage unit.
My entire loose collection is now stored in the 56 quart Sterilite tubs that you get at Target for $5. I use the Target "RE" ones, instead of the ones sold by Sterilite. They're both made by Sterilite but the Target ones are cheaper. Seems silly to me. If you watch after holidays such as Easter, Christmas, Halloween, etc ... you can usually find your local Target blowing them out for $3.50 a tub which is really cheap.
Each of my figures are stored in ziploc freezer bags (quart or gallon depending on the size of the figure). I store the accessories in ziploc snack bags and place them inside with the figure in the larger ziploc bag. Each tub is sorted by figures from a particular series.
I use the extremely awesome shelving units that you can by at Target also ... they're also the "RE" branded ones. You can't find shelving units this inexpensive at any other store. I don't know why Target sells them for so much cheaper than everyone else. Menards might sell one that is comparable in price (within $10), but the ones at other places are always considerably more expensive (somewhere around $100).
The tubs won't protect the figures from extremes in temperature. Storing them like that is doing so at your own risk.