We have "builder's choice" cabinets. Want to add mesh. How?

Not sure exactly what you are asking but maybe this video is a start? https://www.youtube.com/watch?v=-6HqxKZzLk4

Here is how I removed my center panel in cabinet doors to put up wire. I drilled holes into the panel area at each corner. The hole had to be bigger then my jigsaw blade. I did this to allow me to turn the corners with my jigsaw. After panel was removed I used sand paper to smooth down. Then I put up the wire using electric staple gun. Hope this helps. Was pretty easy.

Couple of cautions. The stiles and rails on your doors are somewhat narrower than those on the door in the video. Typical of "builder grade" cabinets. So when you get that panel completely out there is not much holding the frame together. Typically on that type of cabinet a rather small amount of glue. So be aware in may fall apart and you will have to find a way to put it back together. Often on that type door it is the plywood panel mostly holding it together. One solution is neatly cut the center of that panel out leaving the structure of the door intact. Leave maybe half or three quarters of an inch showing. Sand and stain the resulting edge and install your mesh. You might be fine with the video. Just be careful if you want to save the door. I've been a custom cabinet maker for a lot of years and done a lot of this sort of thing. There are a lot of corners cut on some inexpensive cabinetry and you can sometimes take a couple of pieces apart and find yourself with a bunch of kindling.

I agree, your door frames probably would not even hold up to the jig saw, let alone a router!

Thanks. Figures!

It would take more work, and more money, but perhaps build new doors? It's one option, but probably not the one you want.

Since the doors are probably not able to be cut out why not removed them completely and paint the back of the cabinets in a pretty color to add some color to the open cabinet look.