I'm not sure when that would happen exactly, it might just be a vague threat to encourage you to pay, though obviously they've had no problem deleting the board before.
So the options are:
Stay as is, but risk the board going down randomly at any time.
Pay $10 a month to keep it up. This isn't a massive amount especially if it's split between at least a couple of people, but also feels kind of ridiculous for an archaic forum that's been free for the last two decades.
Find another free host. Surprisingly hard to do! Boardhost is the only one I've found that does this style of board with any level of professionalism.
Make our own version and host it somewhere. I kind of like this idea because it seems the best way of achieving the goal of keeping Music Babble online until the heat death of the universe. But it would involve figuring out how to make it.
I just came across this ancient script that looks just like the board:
The script itself is here:
Any web designy people know how to turn that into a functioning website? I assume the babble HMTL for the headers etc could just be copied in.