Probá de esta manera:
Código PHP:
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
<title>Documento sin título</title>
<script>
function verificar(o){
var s=o.value.split('\n').length;
o.rows=(o.value.length/o.cols)+s;
}
onload=function(){
setInterval(function(){verificar(document.getElementById('textarea'));},30);
}
</script>
</head>
<body>
<form id="form1" name="form1" method="post" action="">
<textarea style="overflow-x:hidden;overflow:hidden;" name="textarea" cols="20" rows="5" id="textarea"></textarea>
</form>
</body>
</html>