Earlier quoted context omitted.
With credit cards, you personally do not have much to worry about, since your card issuer holds the ultimate liability for any fraud that occurs. Just be careful to use a credit card (attached to a reversible ledger) and not a debit card (attached to a less-reversible cash account).
Are you sure? I don't know how credit card companies in the US behave, but here in the Netherlands I called up mastercard to ask them whether I am liable for any fraud that occurs if I do something like this (or send credit card info over email, like so many hotels want). The credit card company tells me, yes I am liable for any fraud that occurs, because email and unecrypted text boxes on websites are known to be in…
If it's linked to a savings account and it's a Visa/MC debit card, for example, then it's a different story. The funds are not insured and so if you loose it it's on you.