Best way to make restfull API in Laravel

Laravel handles basic auth himself, the only thing to do is think where you can use the filter (Laravel handles the basic auth with filters), so:

a) In a route:

Route::get('posts/all', array('before' => 'auth.basic', function()
{
    // Only authenticated users may enter...
}));

b) Constructor in the controller (i prefer this):

function __construct() {
    $this->beforeFilter('auth.basic');
}

Also make this adjust if apply for your case, as laravel docs say:

By default, the basic filter will use the email column on the user record when authenticating. If you wish to use another column you may pass the column name as the first parameter to the basic method in your app/filters.php file:

Route::filter('auth.basic', function()
{
    return Auth::basic('username');
});

Basic Auth Docs

EDITED

In your case then maybe you want implement a custom filter with this two methods as basis.

/**
 * Get the credential array for a HTTP Basic request.
 */
function getBasicCredentials(Request $request, $field)
{
    return array($field => $request->getUser(), 'password' => $request->getPassword());
}
/**
 * Get the response for basic authentication.
 *
 * @return \Symfony\Component\HttpFoundation\Response
 */
function getBasicResponse()
{
    $headers = array('WWW-Authenticate' => 'Basic');
    return new Response('Invalid credentials.', 401, $headers);
}

See the default implementation here: