It's important to stay safe online whatever the activity, and provided you play at well known and respected bingo sites, for example like Jackpotjoy, Gala Bingo, Mecca Bingo, or the other bingo sites listed at Penny Bingo then you'll be fine.
Most sites also have "help sections" explaining their privacy and security policies and also the validity of their games (for example how they're auditted to make sure game play is fair). If you're concerned about using your credit card, then you can pay to play at some bingo sites using Paypal, Neteller or Firepay. If you are using your credit card, make sure you're sending your details via a secure server.
With anything online - be vigilant and sensible! Playing bingo online is safe, provided you make sure you're playing at reputable sites and keeping an eye on what you're doing. If you're unsure, check first before giving out any personal details.