I am not sure what your goal is here. You want automatic dehumidification in two spaces with the same dehumidifier? Why not just use two independent dehumidifiers? I suppose the trouble with this is that the basement dehumidifier will over heat the basement with waste heat as they often do. By contrast, the garage dehumidifier will tend to ice up if it is cool enough so you need a refrigeration system that has defrost capability, like a self-defrost freezer.

There is a way to do this off the shelf. It is called a variable refrigerant flow mini split. You can control the temperature and humidity in each space independently. You can move heat between spaces or to and from the outdoor compressor. It will cost you though.

You need to be careful about putting openings, for ducts or otherwise, between the garage space and the basement. There may be specific fire wall regulations for walls between garages and living spaces.

My goal is to get the most use out of a single high quality dehumidifier. They are really expensive. I figured that clever ducting could keep the air in the spaces separated, while letting one heavy duty unit alternate between servicing one room and the the other.

If the rooms are at different temperatures then the humidity levels will not be the same. The hygrometer in the dehumidifier can only monitor one of the rooms. One room will be at the correct humidity level, the other will not. I don't see how you can get this to work unless your dehumidifier has a monitoring system that can overcome this limitation.

If there is no vapor barrier between the rooms, drying out one room will help lower the humidity in the other. It is not ideal but worth a try.

That's a really good point about the hygrometer. So even to mount the dehumidifier remotely, I'll need to put in a remote, hygrometer. I'm glad you mentioned that.

Where is the humidity coming from, outside, the slab, activities in the room...?

How well is the garage sealed?

The garage/shop has well insulated walls and ceiling. There is a single width 8' tall garage door (2" thick and well insulated), but that's certainly the major source of moisture.

The 700 sq ft basement is also well insulated. It has new windows on one side, and no direct door to the outside, just a door to the garage/shop. It is partially heated and cooled by a few vents from the upstairs HVAC system, but not enough to really keep it warm. I'm not sure where humidity comes from for this room. We've got poured cement walls, and floor. It is a finished room with sheetrock walls and tiled floor.