Смог решить как-то так. Может кому пригодится:
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width">
<title>JS Bin</title>
<style>
.active {
color: red;
}
</style>
</head>
<body>
<div id="january">January</div>
<div id="february">February</div>
<div id="march">March</div>
<div id="april">April</div>
<div id="may">May</div>
<div id="june">June</div>
<div id="july">July</div>
<div id="august">August</div>
<div id="september">September</div>
<div id="october">October</div>
<div id="november">November</div>
<div id="december">December</div>
<script>
let date = new Date();
let month = date.toLocaleString('en', { month: 'long' }).toLowerCase();
document.getElementById(month).classList.add("active");
</script>
</body>
</html>
