Updated August 3, 2023
Plastic room cards came into vogue in the 1990s.
Unlike the good old heavy room key, key cards are "blanked" when you check out. Thus, if the key "walks" (and many actual keys and key cards do), the hotel and future guests are at no risk, nor does a locksmith have to cut a new key or change the combination in the lock cylinder so that the old key no longer works.
Key cards are also a more efficient means of keeping track of the physical room inventory.
An urban legend (whose 2003 origins are well-known) posits that a guest's name, address, and credit card information are contained within the magnetic strip, usually on the back of the card, but this has been debunked many times by the hotel and IT industries.
The magnetic strip on most room keys contains the guest's name, room number, arrival and departure dates, and a yes or no command that indicates whether the guest can charge food to his or her room. In addition, key cards are encrypted in such a way that they can't be read by ordinary card readers; most cards are unreadable, as they use proprietary encoders and readers. Also, putting financial data onto a key card would be legally risky for the hotel.
Finally, room locks aren't wired into some sort of mainframe; rather, they're stand-alone battery-powered devices.
Some experts caution that if the front-desk machine that codes your card is linked in some way to the registration computer, it's possible to put any information on the card that is in the computer, including payment information. If you can use your keycard at the hotel to bill something to your room, the information is probably linked somewhere in the system.
That said, payment information isn't contained by the card. Your ID information, when swiped, connects to a point-of-sale server. The transaction info is then routed to the property-management server, where your charge-card information is kept and bills are generated.
Of course, a keycard wiped blank of information can be encoded with stolen personal information, though that's quite a different issue. The same thing could be done with a supermarket loyalty card, a casino players card, or any generic keycard. However, an identity thief would have to already be in possession of your credit or ATM card to load it onto a hotel key card.