I haven't tried it but I know it will work. I've fermented just about every other vegetable in brine and it works really well! Asparagus is a slender veggie, should be a pretty quick waiting period before they're nice and pickled! Generally the thicker the veggie the longer it takes for it to be fermented all the way to the middle.
The amount of salt in your brine can vary quite a bit. A rule of thumb is that for each quart of water, a table spoon of salt makes a 1.8% solution of brine. 3.5% brine is considered to make "half sour" pickles. Old timers used to use 10% brine, and this will preserve stuff for a loooong time but the pickles will be so salty you'll have to soak them in fresh water before you can eat them. 5.5% brine, or about three tablespoons salt per quart of water, will make nice pickles. The saltiness of brine is mostly a matter of taste, but in hot weather, a stronger/saltier brine will keep the food longer. You can always put the jar of fermented veggies in the fridge or other cool spot when you think they're sour enough.
I think pickled radishes are particularly amazing, and I don't really enjoy them non-fermented.
I would definitely add some whole garlic cloves to the asparagus jar/crock. And I might suggest adding some whole rinsed grape leaves? The tannins in them are supposed to help veggies maintain their snap.
Calcium chloride is a kind of salt that allegedly helps stuff stay crispy also. I just ordered some, I haven't had a chance to use it in pickling yet.