There is a lot of misinformation in this thread about storing passwords and localstorage vs cookies.
First thing, local storage is not any less secure than storing a user's password in a cookie. In fact, its actually more secure, because the data in local storage is never sent to the server unless explicitly asked to be sent (cookies are, which is why storing a password in them isn't secure).
That means that you can at least be sure that the password will ONLY be sent via HTTPS.
That being said, for some users the convenience of not having to log in can be very very nice. Your best bet is to use some type of client-side encryption to store the password if you want to be more secure.
Otherwise, think more about the type of information you're handling. What is more important? Usability or security? Is it banking information? What type of damage could a hacker do with the user's password?
There is no secure way to do a 'remember me' either, so the above post is completely wrong. Remember me uses a cookie, which is at best, as secure as local storage.
It comes down to preference and need. Perhaps give users the choice?
The following type of remember me can be implemented using local storage, or a cookie. What is the best way to implement "remember me" for a website?