I was just lecturing about Radon last week. Had I known you were up to your eyeballs in picoCuries, I would have invited you.
So I can't say much about actually dealing with it as a homeowner, but I can tell you what you might be faced with. The good news is that 12 pCi/L is really just over the level of concern. In the dozens and dozens of radon situations I've been involved with, ranges around 8-15 pCi/L were common. I think I saw once something around 20; nothing ever higher. It's really not until you get to 100 that eyebrows are raised and I can't recall ever seeing (or hearing) a case like that.
There's really going to be two ways to deal with it. The expensive way and the really expensive way. The first would involve the application of a sealant on all of your beautiful basement flagstone. All cracks and crevices would be repaired and the sealant applied over the top. Unless things have changed, the sealant is like masonry sealant - thick and ugly. If the basement floor is unfinished (i.e. bare soil) they're going to pour a foundation in and seal that. I've also seen situations where if it's a crawl space they'll put down thick plastic - sort of like a green house, but floor level - to seal the floor. I don't think that's a good solution (personally) if you need to access the basement for maintenance or storage. It's cheaper and likely fine in terms of mitigation, but in terms of use, I wouldn't want it.
Now, if there's already a floor, we get into the really expensive way - installation of a sub-slab suction system (surprisingly safe to google) or a drain tile suction system. Houses (at least in NJ) that were built in the mid to late 1980s had these systems installed as part of the construction process - because it's easier (and cheaper) to put them in when you're building the home. To retro-fit a home from the 1850s is going to require someone that knows what they're doing. IF you don't have a floor and they need to pour one in, I would hope they'd also add the sub-slab suction at this point as well, not just relying on the foundation sealant.
Not knowing anything about the Radon tier your home would be in (or what Radon is like in your part of 'Merica) it's hard for me to say exactly what to expect -- but the levels aren't high enough (in my professional opinion) to warrant concern. It'll just be a matter of mitigation, re-testing and certification that everything is terrific.
What I would urge (if the home is on a well) is that you have the water tested for radionuclide contamination. Again, not knowing anything about your part of the country, I can't say for sure if it's an issue. Even in NJ, only 1/2 the state is at risk for this problem (South Jersey) -- enjoy your water mutants! In addition to radon gas creeping into and concentrating in your basement, it can get into your water supply and then blast you in the face every morning when you take a shower. At that point, you're just breathing in the radiation despite not being in the basement.
They'll probably use expanding foam in either situation to seal the sill space between the house and the foundation (assuming it's wood on stone construction). Really, you're doing everything you can to make the home impenetrable to the radon that is trying to get in.
We have one of those sub-slab systems in our house and a fan running 24/7 to collect and evacuate Radon into the side yard. It's pretty much standard issue in NJ at this point.
) Unfortunately the levels came back at 12 pCi/L. I don't know if that's really high or not but regardless we're going for the mitigation. Lucky for us we requested that the seller agree to paying for mitigation if the results came back above acceptable levels and they accepted.
