The only way your current credit card company can know if you're unemployed is if you tell them. If you're applying for a new card, the company will know because the application form won't show a place of employment. It may be worth calling your card issuer if you're on unemployment because you may be able to either arrange a lower interest rate, defer your payments or lower them.

Credit Score

A dip in your income may cause your credit score to fall slightly. This isn't because the credit bureau knows you've lost your job and are on unemployment benefits. They just notice that your income has fallen. Since this may affect whether you're able to pay your bills on time, your credit score will reflect that. It helps to be on employment benefits for as short a time as possible.