NOT a solution, just a workaround and detailed account of my fruitless efforts...
App installed on iPhone 6s running iOS 10.0.1 (14A403). My badge is also stuck on "1" - does not increment with new messages; does not go away.
Called Airbnb at 855-424-7262; pressed zero, press 2 for other issues; eventually a rep came on the line. Explained this vast problem... unfortunately the Airbn rep insisted it's not a problem on Airbnb's end. Could be true, though clearly they aren't doing anything to help from what I see.
After that call, I deleted the Airbnb app, and reinstalled it - NO change with badge, HOWEVER....
While the notifiation badge was still on 1, I found that in iPhone Settings > Notifications > Airbnb was not listed! (insert scratching head emoji). Likewise, the Airbnb app (from iPhone settings) was not showing a notification option.
Here it gets interesting. I opened the Airbnb app independently, went to it's settings, and guess what? The push notifications was turned off (but the badge was on ???). Hmm. So, turned it on, and was redirected to iPhone's settings where notifications for the Airbnb app now appeared. Turned all off, and back on.
Steps that brought a brief relief of badge, but ultimately did not help:
1) Airbnb App settings, turn on push notifications (will get message, "Push Access Denied"). Select "Settings"
2) Selecting "Notifications" and toggle on. Select preferences for sounds, icon, lock screen, etc
3) I found "Background App Refresh" toggled Off or ON didn't matter either..
4) *** Under iPhone SETTINGS > NOTIFICATIONS > Airbnb > Notifications > "Badge App Icon" ... toggle OFF (I toggle all else ON).