If you're worried about your SSN getting out there, or if you're worried about credit card fraud or identity theft, then don't sign up for any credit card at all. In fact, you probably shouldn't have a regular bank account, either, because they are just as susceptible to the same types of fraud (especially with ATM/check cards).
The point isn't that people shouldn't have access to bank accounts and credit cards. The point is that every additional site that has your personal information is yet another vector through which fraudulent activity can find you. When people say "It's a huge bank, you're safe." that's not a very logical way to look at it. Huge banks represent huge targets and their level of security has proven to be just as bad as the neighborhood tax lady who works out of her home office. Huge banks also suck up huge taxpayer windfalls, the likes of which your neighborhood tax lady can't even fathom.
Here's my practical advice soapbox for people with American financial accounts...
1. Always watch your accounts closely. Check them periodically to make sure nothing weird is coming down the pipe. In some cases you must notify your bank of any suspect behavior in 48 hours or less. In general, it's YOUR responsibility to catch and report fraud, not your bank's.
2. Try to keep as few active accounts as possible. Approach each personal information form with heightened scrutiny. Ask yourself if they really need all that information. If you can't figure out why they need it then don't give it.
3. Close accounts that have little or no activity. In the case of banking or investing accounts this needs to be done carefully, over a period of time, to avoid hurting your credit record. Spread out your closures so that they're not occurring in rapid succession. Also consider requesting a raise in available credit for accounts you intend to remain open to offset the loss of available credit from the account you're closing.
4. If you don't like working under an opaque system that expects you to pick a bank with no way of verifying that they're safe then make those feelings known to your Senators and your Congressperson. Seriously.
5. Review and research any advice you receive online, including in this very post. Don't just blindly follow what this or that person says. Develop a healthy level of personal responsibility and act on it. Take what you read here and elsewhere and go on a fact finding mission on your own to either confirm or discredit it. Then post what you find and let others return the favor.
OK, off my soapbox for a bit. ;-)