Encrypting multiple small messages with a single reused symmetric key
You can use any authenticated, secure cipher. AES-GCM is currently used a lot. You need a nonce per message to use AES-GCM or you can use one of the SIV variants, which calculate a nonce (Synthetic IV) assuming that the messages are all distinct. 

The question is more how to store the symmetric key than anything else, but that's not really a cryptography question - it's more about browser security and offline storage and such. Deriving a key from a passphrase might be a good idea, but note that you need a good key derivation mechanism in addition to a strong passphrase.

Often it makes sense to use asymmetric cryptography so that you don't need the private key during encryption. Keeping the key secure is the main issue with cryptography after all. In that case the passphrase could be used to wrap / unwrap (encrypt / decrypt) a private key value. Unfortunately you'd need local storage to store the wrapped private key - and this might be an issue in itself.