I’m assuming you’re running php artisan serve. Running that should output:

Laravel development server started: <http://127.0.0.1:8000>

Notice how it says the server is accessible through http://127.0.0.1:8000 that is equivalent to http://localhost:8000. Since it’s a development server, Laravel makes it accessible on the localhost loopback interface only, which means that it wouldn’t be accessible from outside of the Droplet as a security measure. That’s why browsing to http://138.68.xxx.xxx:8080/ isn’t working.

Since you have Apache set up, edit the Apache config and make sure DocumentRoot for your domain name is set to /path/to/laravel/public (for example: /var/www/html/laravel/public). Doing so should allow Apache to serve your Laravel application in a production environment, on port 80.