When you log in in your server, in the welcome message you can find your mysql password. This password isn’t the same as your ssh user.

-------------------------------------------------------------------------------------
Thank you for using DigitalOcean's LAMP Application.
[...]
Your MySQL root user's password is password here
You are encouraged to run mysql_secure_installation to ready your server for production [...]

How are you trying to connect to the database? Are you catching the error message? That might give you a better idea of what the problem is. Try doing something like:

try {
    $conn = new PDO('mysql:host=localhost;port=1234;dbname=$dbname', $username, $password);
    $conn->setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION);
} catch(PDOException $e) {
    echo 'ERROR: ' . $e->getMessage();
}

Many thanks both of you. I didn’t execute the sudo mysql_install_db command into my droplet.

Many thanks all your answers.

These a serie of eight videos deeply explaining how to connect to the database using PDO method. Enjoy watching them.
http://curs.io/VvRVYgWV