There are a couple of tricks that photographers use for storing lenses etc. in humid conditions. I suggest they can be usefully applied to razor/ brush care.

First, the cheap and easy idea. We all know those sachets of silica gel that are put in the packaging of electrical goods to keep them dry. They are useful but you can never find them when you you want one. Any that you have wisely set aside for future use will have absorbed enough moisture to match air humidity, so will be of limited use unless you pop them in the oven for a good time to dry them out.

What if you could easily buy tons of this stuff at your local supermarket, not only cheaply, but in an airtight bag and in a very dry state? You can, it's called cat litter - the "pearls" type. It's just silica gel in roughly quarter inch beads. Great stuff, just remember that once you open the bag it's going to start absorbing moisture so keep it stored in an airtight container. Also, it's not magic - whenever you use it to dry something or keep it dry you need to use a sealed container. If it is exposed to open air it will absorb moisture until it matches air humidity, then it becomes pretty much inert, maybe acting to stabilise humidity a bit, but no use for serious drying.

The second idea is to store items in a cabinet with a low power lamp inside near the bottom. This heats up the air in the cabinet slightly. This stops condensation, and in the case of camera lenses, prevents fungus growing on the glass surfaces. Photographers use these a lot in humid countries apparently. It's important to realise that the objective is not to heat the cabinet like an airing cupboard - you are just making the temperature slightly higher than ambient to change the condensation point.

I think an illuminated glass display case holding a collection of nice razors would not only look great, it would help them to dry quickly and stay rust-free. Shaving brushes should also be OK in there, just make sure the lamp is of low enough power to avoid excessive heat building up.

Using this method you could safely store razors in a bathroom cabinet without having to worry too much about humidity.

A glass cabinet might be safer than wood to avoid any fire risk.

I think I will set up a cabinet like this at some point. Although with only 1 straight razor and brush so far I may give my lenses priority in the cabinet at first.