A couple of articles have come to my attention in recent days on the topic of anxiety in the workplace. The proposition put forward in the one I have attached at the end of this post is that anxiety is a symptom of the pace and intensity of modern work - higher expectations to deliver and needing to always be on. Is that your experience? Would you say more employees have anxiety today than before? Now consider this - Is it critical or even advisable for you to determine whether your employee has 'anxiety'? The answer in my opinion is no. What's more important is to identify the symptoms your employee is experiencing and seek their opinion on how those symptoms are influencing them. Whether those symptoms meet a psychological definition of 'anxiety' should not be your focus. Signs of stress that are within the 'normal range' and ones that are significantly impacting on the employee's general functioning are equally worthy of a check in conversation by you as manager.
The question for you is what are you going to say or do in that conversation.
Whether the symptoms meet a clinical definition of anxiety is not critical because your approach shouldn't differ. Your interest in helping your employee exists whether the problem is big or small.
There are two good questions to ask. You can make your own variants of these.
"I've noticed (describe the behaviours you've seen) in recent days, it looks like they are impacting on you a bit?"
This question gives you an opportunity to gauge not only what is going on but more importantly how much distress (if any) your employee is experiencing and what it is impacting on - from the employee's perspective.
The second question is:
"What could we do to help alleviate some of your distress?"
What you are effectively saying here is 'what reasonable adjustments to the work environment can we make that will help you?'
From their it's the employee's call. The employee may not wish to give any indication that there is a problem and/or may minimise its impact. In that event there is little more you can do - you have taken the critical action you need to take. You could advise the employee of various resources that they can access and by all means you should continue to monitor the employee's behaviour and gently repeat the questions on the basis of further observations of the symptoms. But you can't force the employee to do something. If they chose not to engage with you and the symptoms are impacting on performance, the employee needs to be ready for the potential that they may be deemed to be under-performing at an appropriate stage in the future. Hopefully that realisation will encourage them to admit the symptoms and their impacts and work with you to define reasonable adjustments.