持っていません is the more appropriate. Because under this context, it says what needs to be said in the least surprising way.
To my understanding, the cashier only wants to know if you have the card right now and here:
ライフカードを持っていますか？ Do you have a Life [members] card with you? (So you can earn points now?)
In this case, just saying 持っていません or ありません automatically becomes a shorthand for "No, I don't have it with me":
いいえ、[今は]持っていません。 No, I don't have it right now.
いいえ、[今はここには]ありません。 No, it's not here right now.
Since the cashier used the verb 持つ in her question, returning the same verb is the most natural answer here.
If you want to emphasize that you don't even possess a Life card, you need a special marker to widen the context from "here and now." Even then, it would sound awkward or angry to bring it up since it wasn't asked for.
もともと持っていません[し、毎回聞かれてうんざりです]1。 I don't have it in the first place[, so don't bug me about that every time].
 It's not logically implied, but some may interpret it this way.
As for ありません, I can't think of a natural way to express "No, I don't possess a members card" with it.
ある/ありません needs to be used in conjunction with certain categories of things and/or within certain contexts to mean possession. To list a few correct examples,
私には夢がある。 I have a dream. (From the famous speech by Martin Luther King, Jr.)
もしも私にハンマーがあれば、 If I had a hammer, ("If I had a hammer" by Peter, Paul and Mary)
私には夫はありません。 I have no husband. (John 4:17)
私にはカードはありませんが、埋め込みチップならあります。 I don't have a card, but I do have an embedded microchip. (made-up example)
I won't go into details here partly because it'll be out of scope for this question and partly because I don't have a clear answer. If it interests you, here's a lengthy study by someone about the usage of ある.