During the intellectual debate over the Constitution, the Anti-Federalist case against the Federalists' proposed system of checks and balances was made in a number of different ways. It is worth understanding the logic of the Anti-Federalists' arguments before we turn to the Federalist response to those arguments.
A first case made against checks and balances is an obvious one: that it diminishes direct accountability to the people on the part of the government itself. This is, of course, not only a case to be made against the system of checks and balances but a charge to be made against certain features of the Constitution overall: when we ask (for example) what was the intended purpose of the Electoral College, it is precisely this -- that by placing a symbolic entity between the great mass of the electorate and the executive authority that is elected, there might be some way of holding at bay "the people" (conceived, in the worst possible scenario, as a mass or mob held in thrall to demagoguery) and not putting their hands directly on the reins of the executive branch, by interposing an additional step in the electoral process. The Constitution likewise made no provision originally for the direct election of Senators (although it has since been amended to permit this). In both examples, it is perfectly clear what the Anti-Federalists feared about "checks and balances" -- that the country could end up, as it did in 2000, with the electoral college victory of a Presidential candidate who lost the popular vote, or that the Senate would be viewed as an increasingly unrepresentative and undemocratic body.
To argue that checks and balances are a necessity only in societies structured along hereditary lines -- that is to say, with a constitutional monarchy like that of Great Britain -- was similarly an anti-Federalist line of attack against the proposed system of checks and balances in the Constitution. This anti-Federalist objection partakes of the rhetoric of the American Revolution which leaned most heavily on anti-monarchical sentiment: without the hereditary rule of the Hanoverians, in which George III could go mad with porphyria but the British Constitution (such as it is) offered no provision for the replacement of his executive function, save placing his feckless dandy of a son in the position of Prince Regent, there would be no reason to place checks on the executive branch of a government, which (being accountable to an electorate) would not descend into monarchical despotism. But the anti-Federalist line of argument is here absolutely clear if we recall the intellectual traditions in which they were steeped: the fact that the U.S. Constitutional debate was conducted with pseudonyms like "Cato" and "Publius" shows precisely that it was the model of the Roman Republic which suggested itself (viewed perhaps through the enlightened lens of Montesquieu's analysis of Rome's Republican government). Federalist and anti-Federalist alike knew well the story of the gradual collapse of Rome's Republic into a de facto despotism following civil wars between Marius and Sulla (resulting in Sulla's dictatorship) and then Pompey and Caesar (resulting in Caesar's dictatorship and ultimately the establishment of hereditary rule after Caesar's military adjutant Mark Antony was defeated by his political and adoptive heir Octavian, thereafter Caesar Augustus). In other words, for anyone establishing a Republic the lessons of history have shown that a society may very well begin to organize itself along hereditary lines even if it has not done so before.
The subtlest of the anti-Federalist arguments against the principle of checks and balances is to argue that the way in which the Consitution specifically constructs the checks and balances between the governmental branches is, in itself, a violation of the necessary principle of separating the functions of those branches. In other words, there was an anti-Federalist strain of thought which held it vitally important to avoid the concentration of power in the hands of any one particular branch of government, but if -- as in the case of the Federal Judiciary -- one branch is to be appointed to its position for life by another branch (as Supreme Court Justices are appointed by the Executive branch) then that vitiates the argument that, in this case, the Judiciary is itself independent of the Executive. Because the great majority of anti-Federalists were not opposed so much to the idea of checks and balances as they were opposed to the idea of a complex, rather than a simple, structure of government, this argument tended to reduce to an emphasis not on the principle of separation of function so much as a belief that -- despite all arguments made that checks and balances would prevent despotism -- what mattered most in terms of "good" government was responsibility on the part of governmental officials, a thing which could neither be legislated, nor enforced by any particular proposed constitutional structure.
The responses of the Federalists to each of these anti-Federalist objections to the system of checks and balances as it is laid out in the U.S. Constitution are relatively easily stated. The issues of separation of powers, and of checks and balances, are handled at some length in The Federalist Papers, particularly in those papers numbered 47 through 51. (Scholars agree that papers 47 and 48 can be confidently ascribed to James Madison's authorship, while 49 through 51 have their authorship divided variously between Madison and Alexander Hamilton, although considering that Madison and Hamilton, with John Jay, consciously elided their authorship under the joint pseudonym of "Publius," I think we may consider the Federalist position here to be a united one, and not that of any particular individual.) As to how the system of checks and balances might, from an anti-Federalist position, seem to diminish the responsiveness or accountability of the government to the people, this is no problem for Publius. Federalist Paper number 48 quotes at some length Thomas Jefferson's "Notes on the State of Virginia" in order to make the case that concentration of power in the hands of the legislature -- which seems to be the default anti-Federalist solution to the problem of checks and balances -- is a form of despotism like any other. "An elective despotism was not the government we fought for," Jefferson writes in the passage quoted here by Madison, and goes on to say that the government that was fought for was "one which should not only be founded on free principles, but in which the government should be so divided and balanced among several bodies of magistracy, as that no one could transcend their legal limits, without being effectually checked and restrained by the others." In other words, the basic principle is that any government established by the Constitution should be prevented from too easy a slide into despotism, for the overthrow of despotism was the rationale for the Revolution in the first place.
Number 51 of The Federalist Papers lays out the Federalist response to the argument that checks and balances are best suited to a society which has a hereditary aristocracy and monarchy like Britain's. There, Madison (or possibly Hamilton) makes the case on behalf of those hereditary orders, fully aware that Americans are not likely to respect or adopt such a system of titles and prerogatives. But according to Number 51, the potential tyranny of the majority can be forestalled by the existence of such hereditary orders "by creating a will in the community independent of the majority, that is, of the society itself," which -- as Madison notes -- "prevails in all governments possessing an hereditary or self-appointed authority." It is at this point that the Federalist response is summed up, by noting that the sense of checks provided by hereditary orders "at best is but a precarious security, because a power independent of the society…