Suggestions:
Provide more information with $! when open fails (you'll apreciate it when it happens).
open (LOGFILE, "< $logfile") or die ("Could not open $logfile: $!");
You're kind of care free when mixing $thiscust with record fields, hash keys and file handles. Maybe something more like this:
$thiscust = (split /\//, $fields[4])[1];
if ( ! $cust{$thiscust} ) {
print "customer $thiscust\n";
$cust{$thiscust} = $thiscust;
open (my $fh, ">> ${prefix}${thiscust}${suffix}")
or die ("Cannot open ${prefix}${thiscust}${suffix}: $!");
$cust{$thiscust} = $fh;
}
my $fh = $cust{$thiscust};
print $fh $_;