Our old barn was destroyed in a storm. We did the clean up ourselves, stacked the wood and then sold it. While this was a different situation than what you are facing, I still learned a few things that might help.
Taking down old barns is dangerous, so many things can go wrong. So, I would second the advice about protecting yourself in case they would get hurt.
It is VERY hard work. It would be good if the people you have taking the barn down aren't new to the task.
There will be unusable wood/wood they don't want. Whether it be because the wood is rotten or busted or whatever. We burned the junk wood. 100 year old barn wood makes some huge very hot bonfires
So, ask what they plan to do with that....burn it...take it to a dump...make sure you know and approve of what and where and how they plan to deal with what they don't want.
NAILS omg I don't think we will ever get all the nails up. We bought a magnetic roller and that really helped get a bunch of them up. Be aware that they will be a problem. Either ask whoever does it to roll for nails afterward or be prepared to do it yourself. But you'll never get them all lol
What kind of floor does the barn have? If it is just dirt your in great shape. Ours was part dirt, part very thick very broken up concrete, and part logs with wood flooring on top. And there was concrete block fondation (the original rock had been replaced at one time). We needed to rebuild on the site so we had get all that up too. We had to hire a guy with heavy equipment to do that part. They dug a huge hole, put all of it in there and covered it up.
We got a thousand dollars for the wood. Hand hewned beams, barn siding, the log rafters, 4x4 posts. We had it all stacked and seperated. The guys who bought it had torn down lots of old barns, but I don't know how much they paid to do that. It would have been nice to have someone come in and do that, but with that barn being such a blown down mess it was a very dangerous undertaking. I don't know if someone would have come in and cleaned it up for us. Truth be told I'd have given that wood away if someone would have offered to clean up the mess
So, keep that in mind when you are coming up with a price. If you don't want the barn there anymore and they are willing to take it down for you and pay you then I'd say you are a lucky person.