If you are experiencing this issue read on for the official solution from Ubuntu.
Encrypted Home Directory
If you have an encrypted home directory, SSH cannot access your authorized_keys file because it is inside your encrypted home directory and won’t be available until after you are authenticated. Therefore, SSH will default to password authentication.
To solve this, create a folder outside your home named /etc/ssh/<username> (replace “<username>” with your actual username). This directory should have 755 permissions and be owned by the user. Move the authorized_keys file into it. The authorized_keys file should have 644 permissions and be owned by the user.
Then edit your /etc/ssh/sshd_config and add:
AuthorizedKeysFile /etc/ssh/%u/authorized_keys
Finally, restart ssh with:
sudo service ssh restart
The next time you connect with SSH you should not have to enter your password.
Credits: https://help.ubuntu.com/community/SSH/OpenSSH/Keys#Troubleshooting
For a more in-depth discussion we recommend having a look at the following Unix StackExchange entry.