This is unfortunate, but was easily avoidable. If you sign up for a site like this, take reasonable precautions.
This means an alternate email account (and obviously a fake name.) If you actually subscribe, use a prepaid debit card ("gift card") not directly linked to your actual name or bank account.
As someone who's done this (not for ashleymadison), it's not necessarily 'easy'. Also, many sites that take recurring payments won't allow you to use a prepaid card.
I've done it for some other sites, though not recurring though. My guess is if there's enough balance on the prepaid card (for a pre-auth?), it will work.
I never used AM, but I assume they authorized payments on a subscription (recurring) basis, like every other dating site. If that's the case, the kind of pre-auth is different, and one which prepaid cards would reject.
This means an alternate email account (and obviously a fake name.) If you actually subscribe, use a prepaid debit card ("gift card") not directly linked to your actual name or bank account.