Madman, Mercenary, Spoils, and each of the Prizes cost $0* instead of $0, presumably to remind players that they are not part of the Supply (in addition to the reminder in the card text).
Peddler costs $8* instead of $8 to remind players that its cost changes all the time (in addition to the reminder in the card text).
Shelters cost $1, but are never part of the Supply (but have no reminder in the card text).
Is it just because the rules for Shelters specify that they only ever are in players' starting decks? I'm not completely satisfied by this answer; the rules specify for each of the $0* cards that they are not in the Supply (and they even get a reminder on the card text! Why don't Shelters have this? There's plenty of room, except possibly on Overgrown Estate).
Giving Madman, Mercenary, and Spoils the * makes sense, since otherwise each of them looks a lot like a normal Kingdom pile; there's 10 cards in the Madman and Mercenary piles (just like other Kingdom piles) and 15 Spoils (Rats is a Kingdom pile that breaks the rule of 10 cards, so why not Spoils?), but Prizes and Shelters don't closely resemble any other pile.
Basically, it looks to me like cards have * next to their cost if they have a variable cost (Peddler) or if they're not part of the Supply. Once in your deck, the * doesn't matter (except for Peddler--costs $8 for trash-for-benefit stuff, but still matters that its cost adjusts in games with Farmland) since you treat cards that cost $0* as if they cost $0. So since Shelters are never in the Supply, I think they should cost $1*.
One possible reason I can think of is because $1* might be a confusing cost for new players, but considering other cards in Dark Ages I don't think that argument holds any weight. Another explanation I can think of, and I think is more likely, is that no one thought of making them cost $1*. Third, and probably most likely, is that their special type and color should be enough of a reminder that they are not part of the Supply. (but Prize is a special type and it still gets a reminder...)
Isn't nitpicking fun?
tl;dr The costs $1 and $1* are totally different! And Shelters should cost $1* because they're not in the Supply. But maybe the special color makes up for the lack of * and reminder text on the card.