Your Social Security number is a unique Identifier. It may be the most important of what is known as "personal identifying information".
Don't ever use or display your Social Security number in an unencrypted email. (Encryption is a method by which you can't protect any information but especially confidential information transmitted over the internet. You can find applications or technology products to help you with encryption.)
Don't ever attach a document which contains your Social Security number or any financial account number to an email unless both email and the attachment are encrypted.
Don't ever give your Social Security number to a vendor or a caller who contacts you. That caution probably also applies to any bank, credit card or financial information.
You shouldn't give information to anyone who initiates contact with you. You may have a bit more confidence if you are contacting some company or institution and are requested to provide personal identifying information, but even then be careful to provide only information that is necessary to persons with a real 'need to know."
Protect your personal identifying information and your Social Security number as if they were dollar bills in your wallet.