I suddenly started having this problem, too. No text messages for bookings or inquiries. I do get an email. It started about the end of February 2019. My wife's isn't working either. She has a smartphone with the app and I have a "dumb" phone and no app so that's not the problem. I tried deleting my number and readding it, unchecking & checking the SMS box. Nothing works.
I called Airbnb support. Here's the email response I got a couple days later, "I just wanted to follow up as I did hear back from the IT department. They are aware of the issue and are currently working to resolve it. They did not give a time frame of when this will be fixed, but they are in the process of rectifying the problem."