"I have kids your age" is not really something you want to hear from one of your direct reports. However, as the workforce becomes less tenure-based and more results-based, this is going to become familiar to a lot of managers.
An employee who keeps bringing up the fact that they are older than you or that they have more experience than you is conveying several different things. One, they might be confused because their impression was that tenure and experience was most important, and therefore it doesn't make sense why you would be considered higher-level than them. I've met employees who genuinely don't understand why twenty years of doing pretty much the exact same work isn't valued more than ten years of progressively more complex work. Some of this is due to organizations having a culture of promoting people based on a schedule rather than merit, and some of it is because people often don't know what they don't know.
Two, they might worry that, due to their age, you won't value what they have to contribute. It might be tempting to just laugh it off, but if age keeps coming up in team conversations, it might be worth addressing the issue head-on.
Identify the employee's strengths, and find ways to both use and acknowledge those strengths.
If nothing else, they likely have deep institutional and industry knowledge. Don't patronize them or be fake about it, but if there is an opportunity to consult them based on their experience, it's worth going a little bit out of your way to do so.
Focus on results.
When setting goals or communicating in general with the employee about their work, be as specific and measurable as you can. This helps to get the focus away from age and qualifications to actual results.
Find ways to gently convey opportunities or gaps.
This is tricky, because the employee may view you as the one who needs to prove yourself, rather than vice versa. However, the previous tip on focusing on results will help here. Set clear enough targets that it will be clear to the employee if/when they're not measuring up.
It's also helpful to find ways to establish specific things that they will need to do or demonstrate to be considered for higher levels of authority or position. Sometimes managers are afraid to say anything because they value the employee's experience and don't want to hurt their feelings by pointing out deficiencies, but if the employee genuinely doesn't know why they're not advancing, withholding that information from them is unkind.
Hold them accountable.
Yes, they may be older, and yes they know the organization better than you, but that doesn't mean they get to coast. It may sound counterintuitive, but one of the most respectful things you can do as a manager is to hold employees to a higher standard than they're used to. This doesn't mean being unreasonable, and it's important to make sure you provide the necessary information and support to help them get there. Research shows that a sense of accomplishment, especially if it's beyond what the employee previously thought possible, is incredibly motivating.
Explain your reasoning for changing things.
One of the reasons older employees feel disrespected by having a younger manager is that the manager often wants to do things differently than the employee was taught or has been used to doing things. This is common with all managers, but it feels different to the employee when the manager is perceived as knowing less than the employee due to age. It can be helpful to explain the rationale behind your decisions, and to ask the employee to explain the rationale for their way as well. You may find that a combined approach is better, and even if you stick with your original idea, the employee will feel like they have been listened to and treated respectfully.
Be thoughtful about how age differences might affect frames of reference, communication preferences, timing and format of team-building activities, etc.
Diversity can create an enriching team environment with lots of opportunities for mutual learning, and age differences can be part of that as well. Just like with anything else, though, it's important to make sure that no one on the team feels singled out or left out of conversations because of their age difference.
Find areas of commonality between you and the employee and between the other members of the team, too. It's especially important when you first start supervising a much older employee to find things you have in common, both professionally and personally, to help smooth over the inevitable discomfort of getting used to each other.
Recognize that they may be motivated by different things than you.
Not everyone wants to constantly advance in their career. Your employee may have found a role where they can really thrive, or perhaps they like the lifestyle associated with their current role and do not wish to take on more. It's important to not assume that the employee wants to move up or take on more responsibility. This can actually be a good thing - deep expertise and experience in a role can be a wonderful asset to a team, especially when the manager is new to the organization. Take time to get to know the employee and understand what their primary motivations are in their work, and then manage accordingly.
If all else fails, address it head-on.
Sometimes, an employee just won't let go of the age comments. Even if the comments are presented as jokes, it's not appropriate for the employee to keep emphasizing age (of their own or anyone else on the team). Depending on the circumstances, you can either address it in the moment or wait until your one-on-one meeting with the employee. Either way, it's important to be clear that the age difference isn't relevant and shouldn't continue to be a subject of conversation.