If you really want to keep a user logged in then you should store a unique login token in a cookie, not the username or hashed password. When your server receives a login token you can compare it with your database of persistently-logged-in-users and see which user the token corresponds to. The server should store the login token, the username associated with that token, and an expiry time. Once a token has been used it should be invalidated so it cannot be reused.
This minimizes information leaked to the client, and it limits the opportunities for an attacker to recover a legitimate user's password if they manage to steal the cookie (which is relatively easy to do).
I highly recommend that you also set the secure flag on all your cookies so they are only sent across secure connections, and make sure you have a relatively short timeout on persistent logins. Also, it's a good idea to have additional authorization checks, such as making sure the login token is associated with a particular IP address or browser fingerprint, to help prevent casual attacks. This still won't seriously hinder a determined attacker but might dissuade some script kiddies.
Finally, please consider taking @Craig T's advice and only remember the username, rather than keeping a user logged in. Persistent logins are very dangerous so you should think carefully about the value your users get from this vs the potential costs.
Good on you for correctly storing your passwords in your DB! It's amazing how many people think they don't need salts.