This code works.

use LWP::UserAgent;

my $url = "http://prism.library.cornell.edu/control/authBasic/authTest
+/";
my ($servername) = $url =~ m!://([^/]+)!;
my $port = 80;
my $realm = 'User: test Pass:';
my $username = 'test';
my $password = 'this';

my $lwp = LWP::UserAgent->new( );
$lwp->cookie_jar( {} );
$lwp->credentials("$servername:$port", $realm, $username, $password);
my $response = $lwp->get($url);
print $response->as_string;
[download]

In your code you have:

my $url="$url";
my $req = $browser->get($url);  <--- response object, not request obje
+ct
$req->authorization_basic($username, $password); <--- here is where yo
+u pass your response object
                                                 <--- generating the e
+rror.....
my $res = $browser->request($req);
print $res->as_string;
[download]