IMO, you need to use strictures, after which the test for empty input should be performed first ... and then go on to process the entered data e.g.

#!/usr/bin/perl

use warnings;
use strict;

local ($bak_id)=shift;
local ($acc_bal)=shift;

$bak_id=<STDIN>;
$acc_bal=<STDIN>;
chomp ($bak_id);

print "Returning to MM\n", exit 0 unless $bak_id;
.
.
.
[download]

Even better, IMO, would be to use a hash-based dispatch table keyed on (what was previously) $bak_id expressed as a regex e.g.

#!/usr/bin/perl

use warnings;
use strict;

my %dispatch = (
    ''     => sub { print "Returning" ; exit },
    'BOTH' => sub {
        my $acc_bal = shift;
        .
        if ($acc_bal < 1.0) {
            .
            .
        } else {
            .
            .
        }
    },
    'Cust1|Cust2' => sub {
        my $acc_bal = shift;
        .
        if ($acc_bal < 1.0) {
            .
            .
        } else {
            .
            .
        }
    },
);

my ($bak_id, $acc_bal);

while (1) {
    chomp($acc_bal = <STDIN>);

    foreach my $re (keys %dispatch) {
        next unless $acc_bal =~ /^(?:$re)$/;
        &{ $dispatch{$_} }(chomp <STDIN>);
    }
}
.
.
[download]

or some such.

Afterthought: I've only just realised that you don't allow a value of 0 - it's either less or grater than 0 ?