Asymetric cryptography is very secure, websites need methods that are more secured than passwords, so why not using blocchain for that ? 

Authentication

Have u already seen metamask sign-message function ? It allows signing a message without transmitting it to the blockchain. Then, the website uses ur public key for checking the signature. It would also allow using multiple keys for one account, like tron or eos permission system (that allows using an account from another key if permission was set).

Encryption

Another side of ECDSA is the encryption. We can encrypt data using public key, and then decrypt them using private one (it's used by SSH)

The benefit over standard encryption is that a key can be generated without any cert authority, but that blockchain can be used for avoiding phishing (for example, by using an on-chain reputation system)