Is Sierra’s keychain password prompt is annoying the hell out of you, here’s a simple way to fix it. Such a problem can usually arise when your Mac OS user account password does not match your login keychain. There’s two ways to fix this problem.
How to fix problem of Sierra repeatedly asking for Keychain password?
Fixing by resetting Keychain password
The password problem usually pops up after your Mac OS password has been reset for whatever reason. If you have forgotten your Mac OS password, you can reset your keychain password by doing the following. Go to Applications -> Utilities -> Keychain Access. Once there, go to preferences and select option to reset my default Keychain option. Once you enter a new password, Keychain access should create a Keychain that will work with no password.
In some instances, you might not see the option to reset my default Keychain. If that is the case, close the preferences window and select Login from the left sidebar menu of the Keychain Access Window. Press delete and then click on delete preferences.
Fixing by entering old password for Keychain
Go to Applications, utilities and then Keychain Access. Click on Edit and go to the option that allows you to change the password for Keychain Login. Enter your old password that you were using prior to resetting your password for whatever reason. Once again enter this password in the field that says verify password. Click on OK and then quit Keychain access.
Your problem should have ideally gone away by now. If it persists, try a restart for good measure and maybe run the above steps again, in case you made the mistake of making the changes but not saving them.