Thanks, your Excel example helped to get me the answer! The proper syntax is $ftp->Option(AutoCloseMethod), as in:

use Win32::OLE qw(in with);
use Win32::OLE::Const 'CuteFTPPro';
my $ftp = Win32::OLE->new('CuteFTPPro.TEConnection', 'Close');

print "autoclosemethod is currently ", $ftp->Option(AutoCloseMethod), 
+"!\n";
$ftp->SetProperty('Option', "AutoCloseMethod", 1);
print "autoclosemethod is now ", $ftp->Option(AutoCloseMethod), "!\n";
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That got me the output I needed:

D:\>perl -w test.pl
autoclosemethod is currently 0!
autoclosemethod is now 1!
[download]

Thanks for the quick help!