It didn't take long for victims of the Anthem hack to start experiencing problems.

Anthem warned consumers Friday to be aware of scam email campaigns targeting current and former customers of the insurance company. The emails encourage recipients to click a link for credit monitoring services. "These emails are NOT from Anthem," the insurance company said in the statement. Nor, it said, is Anthem calling members about the attack.

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The insurance company announced Wednesday that its database had been hacked, exposing personal data on as many as 80 million records for current and former customers and employees. The information accessed includes names, birthdays, Social Security numbers, street addresses, email addresses and employment information, including income data, according to the company. There is currently no evidence that financial or medical information was compromised, Kristin Binns, a vice president for Anthem, said in a statement.

But there's still a treasure trove of information, Tom Gorup, security operations center manager for Rook Security, told CNBC.com earlier this week. It's enough to commit identity theft, or bypass security questions to lock you out of existing accounts. And the risk isn't short term, like when a credit card number is stolen. "Just because the attacker stole the data today doesn't mean they'll sell it tomorrow," he said. "They could sit on this information for years."

Read MoreWill credit monitoring catch medical ID theft?