This is completely absurd, as detailed extensively on the snopes site shown above. Quite simply, the hotel has no reason to put this info on the stripe, nor in most cases the ability (the card activator is separate from the computer).

How did this urban legend arise? Because criminals decided to use hotel cards as "blanks" to store stolen credit card info that had been pilfered from other sources. Hotel cards are cheap and easy to obtain. When some cops busted the perps, they found the cards with the info on them. And thus the legend began...

Better safe than sorry? Sure. And you can shred every piece of mail you receive, too. But it won't make you safer.