Hi,
It seems it's now a School for the Deaf and there is some information on this site
http://www.sussexdeafhistory.org.uk/page_id__132_path__0p61p56p57p.aspxBasinstoke Heritage Society is also doing some research on the home after being approached by Barchester Healthcare......if I've read the info on the Heritage Society webpage correctly.
It's mentioned under "other matter" on this site, perhaps if you contacted them they may be able to help you.
http://www.bas-herit-soc.org/february_2013.htmlTry googling St Thomas Home Darlington Rd 1874 and you may find a little more.
I have a great love for Basingstoke......when it was a small market town, as opposed to present day. I found the home on google maps and took a little walk around the steets...which seem to have changed from my memory of the area.