Clinton framed the notion of "the establishment" as consisting solely of political bodies of elected officials. Sanders simply argued that a better indicator of belonging to the establishment is one's power and influence within political circles.
As part of the "two for the price of one" that Bill Clinton promised during his rise to the Presidency, Hillary is forced to hide from her role in the creation of the neoliberal New Democrats, the dominant faction of the party. During their joint reign in the White House, the Clintons steered the party far to the right with their draconian criminal justice measures, assault on welfare, liberalization of trade, and deregulation of banking. Their cronies continue to staff the highest ranks of the party and the Obama administration.
Clinton, in a desperate piece of deflection, resorted to playing the gender card: "Senator Sanders is the only person who I think would characterize me, a woman running to be the first woman president, as exemplifying the establishment." This fatuous identity politics is meant to distract from her decades-long tenure at the top of the political system and collusion with those who exercise control over it. Of course, as Bernie points out, Hillary most represents and enjoys the support of the Democratic faction of the political establishment.
But framing the issue as simply a matter of party politics and the electoral system misses the point. Elected officials are merely the public face of the ruling establishment. The broader establishment is composed of the elite class that determines economic policy.
There is no building that says "Establishment" on the door, but there is a century-old institution made up of wealthy and influential representatives of business, Wall Street, corporate law, academia and government. It is a creation of the elite ruling class to ensure their control over shaping policy for their own benefit. Their decisions result in funneling money - and, hence, power - into the hands of a small percentage of capitalists who exercise control over the political process in a positive feedback loop.
In their book Imperial Brain Trust, Laurence Shoup and William Minter write that: "The Council on Foreign Relations is a key part of a network of people and institutions usually referred to by friendly observers as 'the establishment.' " 
The Council was founded after World War I in response to growing domestic social tensions and labor unrest. Socialism was gaining in popularity among the American public in an economic environment marred by exploitative working conditions and skyrocketing inequality.
The Council's mission was to carry out long-term planning for a national agenda. The agenda was meant to undermine a domestic-oriented program that would involve collective decision making to achieve self-sufficiency, and thereby reduce the country's dependence on foreign resources, trade, and other governments.
Some of the many multinationals that subscribed to the CFR's Corporation Service included General Motors, Exxon, Ford, Mobil, United States Steel, Texaco, First National City Bank and IBM. 
"The Council, dominated by corporate leaders, saw expansion of American trade, investment, and population as the solution to domestic problems. It thought in terms of preservation of the status quo at home, and this involved overseas expansion," Shoup and Minter write. 
This imperialist agenda was achieved through manufacturing the consent of the masses (what they called "public enlightenment"), as well as developing foreign policies and ensuring government officials supported and executed these policies.
The Council has been remarkably successful in its mission. It has achieved a monopoly over foreign policy planning, and become thoroughly integrated with the government that carries out policy prescriptions. Entire administrations have drawn their foreign policy officials from the ranks of the Council. There is a steady two-way flow of personnel between the Council and government.
Both Bill and Chelsea are current members of the CFR. While Hillary herself is not a member, she is no doubt influenced by her immediate family's ties to the Council. Additionally, she collaborated closely with the Council while she served as Secretary of State, as she made clear in a 2009 speech at the Council's office in Washington:
"I am delighted to be here in these new headquarters. I have been often to, I guess, the mothership in New York City. But it's good to have an outpost of the Council right here down the street from the State Department. We get a lot of advice from the Council, so this will mean I won't have as hard a go to be told what we should be doing, and how we should think about the future."One of many people whose career was launched by his association with the Council was Henry Kissinger. In the late 1950s, he was appointed the director of a study group on nuclear weapons, in collaboration with several of the Council's directors. The result was a book authored by Kissinger, Nuclear Weapons and Foreign Policy.
Kissinger went on to serve as possibly the most influential foreign policy official in American history under Richard Nixon (and later Gerald Ford), as both Secretary of State and National Security Adviser. He helped carry out war crimes when he transmitted President Nixon's order "anything that flies on anything that moves" to General Alexander Haig, directing a massive, secret bombing campaign of Cambodia hidden from Congress and the American public.
Kissinger's tenure also saw him intimately involved with the military coup led by General Pinochet to overthrow and kill democratically-elected President Salvador Allende in Chile in 1973; the invasion by Indonesia of East Timor in 1975 and the subsequent genocide against the native East Timorese; the South African invasion of Angola in 1975 and attempted installation of a puppet ruler amenable to the apartheid regime; and the Dirty War in Argentina in which leftist opposition members were killed an disappeared.
Rather than being subjected to prosecution, or even suffering a loss of prestige, Kissinger has seen his reputation rise in the decades following his genocidal actions.
Clinton wrote that "Kissinger is a friend, and I relied on his counsel when I served as secretary of state." She noted that they share "a belief in the indispensability of continued American leadership in service of a just and liberal order."
Clinton's abstract and idealistic rhetoric exemplifies the bipartisan, imperialist agenda formulated and propagated by the Council on Foreign Relations. The humanitarianism is a guise for the ruthless pursuit of United States political and economic hegemony across the world. The people who belong to this elite club have internalized the imperialist worldview that the U.S. is an "indispensable nation" that upholds "a just and liberal" world order, and use this belief to rationalize their Machiavellian exertions of power abroad.
The American establishment that matters most is not limited to any one party, gender, or government organization. It is limited to people who are involved, directly or peripherally, in formulating and carrying out the plans of a tiny elite class - plans that ignore the 99 percent of the Americans in whose names they act, and the billions of people whose lives their decisions impact. There is no one whose social relationships and professional career typifies this more than Hillary Rodham Clinton.
 Shoup, Laurence H. and William Minter. Imperial Brain Trust: The Council on Foreign Relations & United States Foreign Policy. Lincoln, NE: Authors Choice Press, 1977/2004. (pg. 9)
 Ibid. (pg. 50)
 Ibid. (pg. 23)