Is the back section of this house salvageable or should it be knocked down?
Rick, that is an impossible question to answer with the info you have supplied. The condition may be bad, but you need to access or have accessed the structural integrity of the shell. This doesn't sound like it's gonna be a DIY project, so my advise is to get some professional evaluation and a few quotes for a restoration and the demolition. Once you know what the real costs are going to be, it will be much easier to make a decision to repair or remove with your family.  If you want to post some pics, it might make for some interesting comments.

After looking at the pics, my first impression is that the frame looks fair and built with decent sized lumber. It looks fairly straight on the back of the building. The is obviously a lot of water damage and the plywood floor has rotted as well as some of the framing in that area. I imagine the roof is shot as well.  As a contractor, I look at it and see a fairly involved renovation. Now depending on what that space is used for and what it adds to the overall property would be part of the considerations. It obviously is gonna need some framing, new floors, roof, windows and siding.  There will be some electrical work as well, I'm sure.  I also see a fairly expensive demo job there. It is a large structure with a lot of wasted to get rid of.  

If this space was important to the overall value of the property, I would consider a rehab. Like I said before, get a competent contractor or two over there and get some quotes.  In my distant and still uninformed opinion, it looks like it might be worth saving.