You can use File::Tee to capture standard output to a file as well as letting it go to the screen. Also, you need to pass the filehandle to rls with a Filehandle hash key:

use strict;
use warnings;
use Net::FTP::Recursive;
use File::Tee qw(tee);
tee(*STDOUT, '>', 'capture.txt');

my $ftp = Net::FTP::Recursive->new("ftp.mozilla.org", Debug => 0);
$ftp->login("anonymous",'me@here.there');
$ftp->rls( Filehandle => *STDOUT);
$ftp->quit;
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