Arkham Asylum

Essential in a deck using Lunatics. Otherwise, it is more of a meta choice where you are being hit by insanity repeatedly, or lack terror/willpower to protect yourself from insanity.

The fact it not only restores but also readies the character is very powerful, as with Lunatics it lets you use their ability, then still allow you to go into stories.

Arkham Asylum or Moon Worshiper. Which one is best?

Within Hastur, I prefer Moon Worshipper.  Outside of Hastur, I prefer Arkham Asylum.

Arkham Asylum lets you restore multiple targets in the same turn, that can be very handy in Hastur decks that drive their own characters insane.

My only Hastur deck that plays with either of these, plays with both of them.

(adding the Moon Worshiper greatly improved consistency, over just running the Asylum)

The dykes of Ys is as far as I know a key card for Hastur. Only Arkham Asylum can restore cards that would first enter play insane.

Also, Moon worshiper only restores a Hastur character that was just made insane. Arkham Asylum both restores and readies.

The fact that Moon Worshipper only restores can be a good thing though when you're talking about a card like Benjamin Herriot.  The Asylum wouldn't go well with Herriot, but the Moon Worshipper does.  Also, the Moon Worshipper is free to play and to use, whereas the Asylum requires initial and repeated investment of resources.  Finally, when you're running heavy on Lunatics, I feel like restoring without readying isn't much of a problem.  You're probably just going to drive your Lunatic insane again anyway.

Both the Moon Worshipper and Arkham Asylum are good.  Like Richard Plunkett said above, use both and enjoy the consistency and flexibility they offer.