... The average decline in membership has been 1,179 per year, and if this continued there would be no members in 11 years time.
What then happens to the buildings and the contents of the library and the museum?
Under the assumption you are making that the trend will continue, what will happen depends on what the income of the organization is at that time.
I have been a member since 1971 and I have no intention of dropping my membership. I believe there are at least a few thousand who feel the same and we represent a "core" membership.
Our current membership is about the same as it was when I joined in 1971. The growth spurt that took us from 14,000 to 38,000 was largely due to a burst in commercial activity in what had been a somewhat cloistered group of collectors and a relatively small number of specialists in horology. The organization did away with a requirement for new members to be sponsored and approved by the membership and Roy Ehrhardt and others began a campaign across all the flea markets in the U.S. to organize the "antique" watch & clock collectors.
Perhaps a little drunk with success, the organization undertook to expand the small telephone building that housed headquarters into the rather impressive museum structure we have today. They also bought the state licensed beer distributor across the street to build a school of horology. During this period, there were actually two organizations. The National Watch & Clock Museum was governed by a self appointed Board of Trustees and the NAWCC Board of Directors provided policy and vision for the membership organization and supervised the Executive Director.
At the end of the 20th century, we were targeted for review by the local IRS office. There was a growing backlash to the cost over runs of the museum building project. eBay appeared on the scene and the commercial promise of the NAWCC Mart room began to dissipate.
The organizations responded to this sequence of events by merging the NAWCC and the NWCM, consolidating the governance, giving the Executive Director a formal Delegation of Authority, and initiating a major volunteer based effort to grow on the Internet.
From my perspective, it seems like we have now brought the NAWCC back to where it was at about the time I joined but with much better communications and a better understanding of what we are about and why people might want to support what we, as an organization, do. There is still a lot of confusion about us being a trader's club or being an educational charity like any distributed on-line university with our fine museum and research center as our anchor point.
Our current development campaign is bringing this dichotomy into sharper focus. I hope and trust we will be able get through this phase and develop a community of people who love horology more than they do making a profit while still embracing a healthy commerce in horological artifacts and information.
In examining trends, the other trend worth considering is the growth of donor level memberships and donations in general. The NAWCC has not been publishing that information as long as we have the membership statistics, but I think it is important in understanding where we are now and where we may be going.
In direct answer to your "what if" question, some other organization with similar goals to ours would inherit our remainder assets after all debts had been paid. As I said, I do not expect that to happen.