|It appears that your employee does not have an open door or an open mind on this matter, and both need to be opened.
Her comment that she is more productive with the door closed is questionable at best. The most effective and productive managers are visible and accessible to their team, and they maintain a high degree of contact and communication with them. She needs to open her door and hear this from you.
If there is a great deal of noise outside of her office, perhaps the employees in adjacent areas should be advised to exercise more volume control. Taking a more sinister look at her behavior, one has to wonder what she is really doing in her office. There are plenty of employees who use a closed door to close out people who might interrupt their personal calls and surfing activities.
You should advise her that she needs to keep the door open, and you will work with her regarding her concerns about this. Since having an open door is part of what it means to be a manager in your company, she will need one if she means to be a manager in your company.