Social Security numbers were never designed to be secure or used as a form of national ID. Their use as such has resulted in a world where Social Security numbers are "for sale online for cheaper than a cup of coffee":
Only a very narrow set of government agencies and financial organizations are required to ask for a customer’s SSN by law. That list does not include landlords, cable companies, cell phone providers, or even credit reporting agencies, which all habitually request SSNs simply because a number is more precise than a name. Americans have repeatedly rejected the idea of instituting a formal national identification system when it was proposed for use in health care or to prevent illegal immigration. As a result, the SSN has become the national ID by default.
Unfortunately, SSNs are also appealing to identity thieves, who can use the numbers to open new bank accounts and credit cards. An SSN is also typically the first piece in building an identity profile that can be used for more elaborate crimes like insurance fraud. In addition to being unique and widely available, the vast majority of SSNs were assigned according to a publicly-available formula.