Culturally responsive teaching can be many things.
Content
The most obvious starting point is looking at the subject you teach. You can find the intercultural angle of whatever subject you are teaching though some are easier than others. Cooking and languages for example include intercultural aspects almost without effort, though you can choose to be more deliberate about noticing when this happens. Typical adult education programmes usually include an extensive range of physical fitness courses such as yoga or swimming and there teachers might consider how to make their classes more inclusive by taking account different levels of modesty. Another popular range of courses usually deal with society and politics and again there, the intercultural component is easy to discern. In personal finance you could consider Islamic banking. In basic IT you could talk about the EU cookie law and why you cannot link to Chinese friends on Facebook. I am wondering if there is any subject that could not potentially include an intercultural component?
Pedagogy/Andragogy
Moving on from subject areas, there is the way that you teach. What is the teacher student dynamic like? What are your expectations as a teacher and what do you assume are the expectations of your students? Could you cope for example with students who have been brought up to defer to authority? How would you organise discussion tasks? How would you draw out their experiences and expertise; one of the keys to success in adult education? And what about feedback? Generally accepted now to be one of the most powerful learning tools, how do you organise and give feedback? Do you praise the person or the group? Do you praise the result or the amount of effort that went into it? And let us not forget that you have your own expertise and beliefs about the best way to promote learning. So how far should you or would you be willing to modify those best practices?
Physical layout
Even the way you organise the teaching room can send powerful messages to your students about the relative status between you and them and the way in which you expect to work.
Best practice
In the end, CRT mainly reflects the growing consensus about what best practice is in pedagogy with all of us trying to create an effective learning community. So a step towards the culturally responsive classroom may be a step towards more effective pedagogy.