How do you currently measure the success of your campaigns when the goal is to target diverse audiences?

Once your audience is defined you need to look at it's estimated reach and then the actual audience reached. The % of audience reached is the starting point then you look at the deeper funnel (i.e. how many users interact , how many users reply, donate etc.) if you don't have benchmarks look at publicly available data for reach metrics. As an example , most email campaigns have a 1% open rate to be considered successful

You will need to benchmark, set cohort and percentage values to diverse groups before the campaign launches. You can also use Google Cloud Vision AI to analyze the visual content of your campaigns to ensure inclusivity.

Afterwards you can use number of tools to help you with this task:
Conduct demographics breakdown via Google Analytics, YT, Meta, this will give you an overview of age, gender, location for those who interacted with your campaign/website.

You can use socialbakers which provides detailed engagement analytics by audience groups, niche or large. To complement this, you can also use Mixpanel to monitor specific user actions and tracks conversion rates by cohort.

For direct customer feedback, you can use Medallia. It captures customer feedback to measure sentiment and satisfaction by audience segment.