Build a hash

#!/usr/bin/perl

use strict;
use warnings;

my %null;

while (<DATA>){
  my @fields = split(/\|/);
  for my $i (0..$#fields){
    $null{"field$i"}++ unless $fields[$i];
  }
}

for my $key (sort keys %null){
  print "$key->$null{$key}\n";
}

__DATA__
first||third||
alpha|beta||delta|
||c|d|
one|||four|
[download]

output:

field0->1
field1->3
field2->2
field3->1
[download]

The field numbers start at zero but also the second field has 3 compared to 2 in your desired ouput.

The third line of your data starts with two bars, isn't that 2 null fields?