What is this white fluff growing in my basement?
Might as well make this an answer.


  ...it seems to collapse when I rub my foot on it.


As @tyersome stated, it appears to be salt, not mold. Salt crystals would definitely collapse easily underfoot, and should be mostly quite dry, whereas mold would be... maybe squishy (?) but likely unpleasant in some way. Mold also smells, which you haven't mentioned, whereas salt is odorless.

To test, scrape a bit up (salt crystals should feel a bit sandy, not mossy) and see what a drop or two of water does. Mold doesn't dissolve in water; it floats. If water seems to dissolve some of it, add more. If it all dissolves, it's probably a salt, and since you live near the beach, it might be mostly salt, i.e. NaCl, with trace minerals thrown in. 

If it dissolves and you're brave, you can taste a tiny bit then spit and rinse. While salt isn't toxic, I can't vouch for what was in the grout.

My guess is that there may be a low spot or two in your basement where salt water is entering through the porous grout; as it evaporates, it leaves salt crystals behind. More water = saltier solution = bigger crystals as the water evaporates. 

If you have a magnifying glass, you should see crystals in the undisturbed deposit. If instead it looks soft and hairy, it is mold. I can't think of a white mold off the top of my head (well, Candida albicans, but that smells yeasty and is definitely not crumbly). Mold/yeast needs a substrate (food) to grow on, and wet grout isn't very nutritious. Mildew, sure, but mot big fluffy mold.

 I don't like this reference because it's not scientific, but it explains efflorescence well.)