There are various ways you could do this. One is to read the entire file into a string and then use split to break it into parts. You should read split carefully, it has many features.

use strict;
use warnings;

my $text = <<EOF;
MAIN
LR R1,R2
.
.
BALR R1,LABEL#1
L R1,VAR1,
.
.
BAL R3,LABEL#2
.
.
EJECT
LABEL#1
.
.
LABEL#2
.
.
EOF

my @parts = split(/^(BAL.*$)/m, $text);

my $n = 1;
open(my $fh, '>', "file$n") or die "file$n: $!";
foreach my $part (@parts) {
    print $fh $part;
    if($part =~ m/^BAL/) {
        $n++;
        open($fh, '>', "file$n") or die "file$n: $!";
    }
}
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