We are doing this. There are many stages in the research process for which AI is excellent. It would be negligent not to teach this. For example, creating search strings in a literature review or coding unstructured data.
My uni is also making quite a bit of money teaching business people how to use genAI in aspects of Decision support. We also support a genAI after-school club for kids within our local area.
I personally feel we will adapt to it quite well and quickly, now that we have got our heads out of the sand. I don't think, for example, it is likely for anyone to get a first-class degree simply by getting genAI to do their work because 1) most good degrees have a variety of assessments in-built 2) the big research databases* have blocked the latest research from the likes of OpenAI. So at best, research using the off-the-shelf AI tools can only find poorly researched publicly available internet resources. So I wouldn't expect, e.g. an MSc dissertation written by copilot to achieve a high mark, though it will probably be quite well structured, etc.
*They are creating their own AI tools, because they are the most greedy of greedy cnuts.