As others have suggested, you could use LWP::UserAgent. This allows you to set the authorization header via HTTP::Headers. You could then transfer the image to a local directory and feed that to Image::Magick.

use LWP::UserAgent;
use Image::Magick;

my $url = 'path/to/remotefile';
my $localfile = 'path/to/localfile';
my $ua = LWP::UserAgent->new;
my $req = HTTP::Request->new;
$req->authorization_basic('username', 'pass');
$ua->get( $url, :content_file => $localfile );
# do some checking here
my $image = Image::Magick->new;
$image->Read($localfile);
[download]

- Untested -