Now that martin has solved your basic problem, I'd just like to point out short your code could be:

my $path = "./mail";
my $bkup = "./backup";

open MAILDIR, $path;

for my $mbox ( grep { -f "$path/$_" and -s _ } readdir MAILDIR )
{
    rename "$path/$mbox", "$bkup/$mbox";
    system( "perl -ne 'print if /^From/..-1' $bkup/$mbox > $path/$mbox
+" );
}
[download]

(That assumes that the backup directory is not on a distinct disk volume.)

update: as capoeiraolly points out below, that version backs up all mailbox files, not just the ones that need fixing. To avoid that, just add a few lines at the top of the for loop:

for my $mbox ( grep { -f "$path/$_" and -s _ } readdir MAILDIR )
{
    my $first = do { open M, "$path/$_"; <M> };
    close M;
    next if ( $first =~ /^From / );

    # ... do rename and system calls on bad files only
    rename "$path/$mbox", "$bkup/$mbox";
    system( "perl -ne 'print if /^From/..-1' $bkup/$mbox > $path/$mbox
+" );
}
[download]