Hi Lisa and Judy (and anyone searching for info about the Daniel Webster Home) -
Yesterday, I found quite a bit of general info about Franklin, NH at their genweb page:
http://www.rootsweb.com/~nhcfrank/

By calling the number listed there for the Franklin Historical Society, I got in touch with a Mr. Albert Garneau, Town Historian for Franklin, NH. He was very friendly and knowledgeable (apparently, he wrote The History of Franklin, Vol. 1!) Mr. Garneau told me that all of the records (and some of the children) were transferred to the Spaulding Youth Center/Foundation in Tilton, NH.
Upon reading about the Spaulding Youth Center here:
http://www.s-y-c.org/home.asp
it seems that it was formed when Daniel Webster and the Golden Rule Farm merged in 1958.
Mr. Garneau gave me a contact number, and I found a few more on the Spaulding site. I think I have found the appropriate person to speak with about the records and I left a message for him yesterday. I plan to wait a day or so, as I'm sure he has a million other things to do besides discuss old records and I don't want to make a pest of myself ... lol. But I will keep at it and will be sure to let any/all of you know how I make out.

Hope this helps ... good luck!