I read somewhere about putting Epsom salt on certain plants, but not regular table salt.... regular table salt kills grass so I would be careful about putting it on vegetables
http://www.garden.org/articles/articles.php?q=show&id=68&page=1 this is a good article "Fertilize with Epsom Salt"
I cannot see how salt water would help asparagus.
Regular table salt is used to kill unwanted plant growth. If you value your asparagus don't salt it until you've picked it and want to eat it!