Asparagus should in fact be refrigerated when you get it home from the supermarket in order to keep it from going bad for as long as possible. Just make sure that you put your asparagus in the vegetable crisper where it will stay its freshest. Since asparagus only lasts for a few days in the fridge, you might want to consider moving it to the freezer if you do not plan on using it for a while.
By getting your asparagus properly stored you will be able to avoid having it lose some of its flavor or nutrition. If you plan on freezing your asparagus you will definitely need to make sure that you either put it in a freezer bag or airtight container so that it is completely and properly preserved.