The Debug=>1 option dumps messages wholesale to STDERR. For what it's worth, something like this will give you the whole session log as a scalar:

use IO::Scalar;

my $stderr;
tie *STDERR, 'IO::Scalar', \$stderr;

my ($address, $timeout, $port) = ('127.0.0.1', 90, 21);
my $ftp=Net::FTP->new($address, Timeout=>$timeout, Port=>$port, Debug=
+>1);

## do ftp stuff..

if ($stderr)    {
    $stderr =~ s/Net::FTP.+?\(.+?\)\n?//g;  # Trim off Debug header, s
+ave meat
    $stderr =~ s/\n+/\n/g;                 # Get rid of blank lines
    open  LOG, '>>', 'some_logfile';
    flock LOG, 2;
    print LOG "$stderr\n";
    close LOG;
}
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The scalar $stderr can be manipulated to create a commented log as you go. e.g.:
$stderr.="Trying to download the file\n"; $ftp->get(...);