No cleaning suggestions - sorry. If you have a leaky basement (most basements remain completely dry regardless of how hard it rains), then the solution becomes: fix the leak.

I know that's not what you are on here for. I get it. The deposits are known as efflorescence. This is when ground water (not rain water) move up THROUGH the slab and into the house. The deposits are "ground salts". They are extremely difficult to remove...even on bare concrete. They take special solutions to emulsify the deposits. Once they are in liquid form (like 1-2mm of standing water on a bare slab) you can then use a shop vac to vacuum up the water (with all the minerals floating around in the water).

So long as you have a leaky basement/porous slab and carpet, you will continue to have these issues. I supposed you could use a wet vac to remove as much water in the carpet as possible....each and every time it rains hard. And then every few months you could use a carpet cleaner/shampoo machine to do a deeper clean. I'm sure a local chemistry student could offer tips on what type of cleaning solution you could use to remove the minerals.

The other thing you will need to monitor: mold and mildew. With a carpet that continues to get wet, you will be increasing the mold/mildew spores in the home. If the basement smells "musty" then you already have a problem. Not all mold is dangerous, but if anyone in your house has breathing/lung issues (asthma, allergies, COPD, emphysema, etc) then the presence of mold spores in a 'musty basement' could be more than they can handle.