It appears that this was coined by George Meany in 1967 to describe federal government programs. The United States Supreme Court is the "court of last resort" and probably where this "last resort" phrase originated.
24 August 1967, Washington Post, pg. A5:
Where other alternatives fail, he said, "the Federal Government must be the landlord of last resort."
(AFL-CIO President George Meany - ed.)
13 September 1967, New York Times, "Labor Urges Plan for Urban Crisis" by Damon Stetson, pg. 1:
"The United States Government has got to be the employer of last resort and the landlord of last resort," George (Pg. 32, col. 7--ed.) Meany, A.F.L.-C.I.O. president, said in explaining the program.
25 October 1967, New York Times, pg. 51:
"The Federal Government must be the employer of last resort and the landlord of last resort," Mr. Meany, president of the American Federation of Labor and Congress of Industrial Organizations, told the Special Advisory Commission on Civil Disorders.
13 February 1979, New York Times, "Officials Debate Financial Ability of City to Play 'Biggest Landlord'" by Arnold H. Lubasch, pg. B6:
But, he said, if the program fails to rehabilitate housing to adequate conditions, "one might suggest that New York City cannot afford to be the landlord of last resort."
(U.S. Representative William Green, Republican of Manhattan - ed.)