If your chemicals are stored in bottles, then they are not going to be significantly affected by any humidity in the laundry room.
Some chemicals are affected by light, and storing them in brown bottles protects them from that form of deterioration.
You didn't mention if you were using any plastic bottles. Theoretically, plastic bottles can breathe, and are not as good as glass for long term storage. Whether this is actually a concern really depends on how long you store the chemicals. If you cycle through a set of chemicals in six months or thereabouts, my experience suggests that plastic is not a problem.
The only other issue is whether you have little people in the house who could get into those chemicals. I know that you said high shelf, but some of those little foks are pretty creative. The best way to store chemicals in a darkroom that is shared with other household functions is to have them in a locked cabinet. And if you do that, then you don't need to worry about brown bottles since the cabinet will be dark most of the time.