No permanent install is needed. Just search for ‘tray divider’ on Amazon. There’s tons of options.

I wouldn't store them under the sink. I wouldn't store any food-contact items or food along with cleaning chemicals or under the sink. That's some food safety 101. Even with a divider, you would still be storing food contact items in an area with food hazards. A mesh door isn't going to mitigate that.

You can put an enclosed box in the cabinet to keep it separate from chemicals. Think shoebox on its side, but larger and wood and nicer.

Mesh fronted cabinets were a thing in the early kitchen days, right around the time skirts on sinks went out of fashion. It can be a really cute look in a traditional kitchen. (The ventilation slits cut into the wood were more common I think.)

There is no "murk" under a properly installed sink.

"murk" is found under sink cabinets that have had a slow undetected leak for years or a major leak that wasn't dried/cleaned properly afterwards. If you smell "murk" something isn't right and should be fixed!

Store things where they will be used. Cutting boards are not usually used at the sink.

Mine are stored in dividers above the fridge which is right across from my prep space.

I keep my large cutting boards under the sink. I bought a basic white divider like Susan L linked. I can get 2 boards in each slot.

Thank you for all the help!

@Susan L - I'll plan to get one of those tray dividers. And for the reasons @darbuka pointed out, I won't nail it into the cabinet base.

@chispa and @Joseph Corlett - The kitchen sinks I've used before (growing up, in various apartments, and in my current rental house) were all murky, so I guess I just assumed sinks got that way. Excited to learn that our kitchen sink in the new house should be clean and dry! I'll scratch the mesh idea.

and @Iri - love the shoebox idea. I'll look into doing something of that sort