Maybe if you printed $line instead of $_? You've faked yourself out it seems.

Also, you're opening INPUT but not actually using it. You're still reading that file because, coincidentally, that is what <> does.

A little touch up:

use strict;

my $count = 0;

open (INPUT, $ARGV[0]) || die "Could not read $ARGV[0]\n";

foreach my $line (<INPUT>)
{
     chomp ($line);
     $line =~ s/(.{60})/$1\n/g;

     $count += />/g;

     print ">$count\n$line\n\n";
}

close (INPUT);
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I'm not entirely sure why you're only printing split lines.