The password prompt is added for additional security on the WordPress admin page. You can disable it by editing
/etc/apache2/apache2.conf From the full tutorial, One-Click Install WordPress on Ubuntu 14.04 with DigitalOcean:
To remove the additional password prompt, you’ll need to log into your VPS instance through SSH.... Once logged in, you need to update the main Apache configuration file. Open it as root with your text editor:
Inside, you need to find the section that deals with the “/wp-admin/” section. It should look like this
AuthName "Please login to your droplet via SSH for login details."
This entire section can be commented out by placing the “#” character before each line, like this:
# AuthType Basic
# AuthName "Please login to your droplet via SSH for login details."
# AuthUserFile /etc/apache2/.htpasswd
# Require valid-user
Once this is complete, you can restart Apache for your new changes to take affect.
service apache2 restart
This should remove the additional password prompt from the administrative areas, which is usually okay since WordPress itself can handle authentication.