I use an Eclipse 900-041 plastic storage box, and keep it in the cabinet under the bathroom sink. The cabinet stays pretty dry, and the box probably also helps in that regard. The box is made for holding small parts, but with the dividers removed it also makes a good blade storage box. Each row can hold three stacks of blades four packs high. An extra box can be stored at an angle at the end of each row. This box also works well for storing loose blades. They sell for for about $4-$7. To find sellers, just cut and paste this into Google: Eclipse 900-041 Plastic Box w/dividers 8 X 5.25 X 1.5 and the search will return a bunch of results. Fry's Electronics has them at http://www.frys.com/product/7300834 and also in some of their stores. The manufacturer, Eclipse, has them at http://shop.eclipsetools.com/IW_Products.m4p.pvx?;PRODUCTS_DTL?cat=900.0700 The compartments are just a little too narrow for my razors, but there are likely similar storage boxes that would work for them, too.
We keep the razors we use in the bathroom, without a problem. We also keep a few packs of blades in the bathroom (in a metal box, in a drawer.) Since the pack is wrapped in plastic, we do not worry about exposure to humidity. As we buy blades in bulk, most of the blades are stored in a tin, in the linen closet.
I have two 4yo boys, one of whom is into knives and swords. I bought a small lockbox for $20 to store my blades and pocketknives when I'm not using them. The nice thing about the lockbox is that if a burglar ever breaks in, he'll think he's made a big score, and when he gets it home he'll discover what he thought was the hail of the century is actually... razorblades