属性require付き:

<select required="required">
<option value=""></option><!-- added empty value -->
<option value="1">01</option>
<option value="2">02</option>
...
<option value="31">31</option>
</select>

requireなし -属性を削除するだけです:

<select>
<option value="1">01</option>
<option value="2">02</option>
...
<option value="31">31</option>
</select>