I'd kindly remind them that cash payments are not accepted on Airbnb and that you only accept bookings through Airbnb, so if they want to reserve, they'll need to use one of the many payment methods that Airbnb accepts. This is for their safety and security, so I can't see why they wouldn't want to do it this way.
Also can share some Airbnb articles with them on the subject:
https://www.airbnb.com/help/article/126/what-methods-of-payment-does-airbnb-accept
https://www.airbnb.com/help/article/199/what-should-i-do-if-someone-asks-me-to-pay-outside-of-the-ai...
If you've pre-approved them, they have 24 hours to complete the reservation. The booking isn't finalized until payment is finalized, so if they stop responding or you don't feel comfortable with it, just a little bit of time and it's done.
You can always Decline their reservation request, too, if you don't feel comfortable with the situation or they don't agree with your rules.