The answer to the first question would be Congress, primarily. As to the second question, I guess the courts would have that burden, for all practical purposes. But I think the real question you were getting at isn't who makes that determination, but how that determination should be made.
I think you're being very charitable by calling suspension of literacy tests a "means" of enforcing anything. It really isn't - it's just a new prohibition. It is simply not necessary, in order to enforce a particular law, to prohibit other activities that might be related. Enforcing drunk-driving laws, for example, does not require outlawing alcoholic beverages altogether. Sure it would make it more convenient, but if the Constitution only gave Congress the power to prohibit drunk driving, by no stretch of the imagination would they also have the power to prohibit alcoholic consumption. They can only prohibit the act itself, not things that might lead people to commit the act. And if Congress, through its investigations, was able to learn that certain states were misusing literacy tests to discriminate against blacks, then they also had the ability to determine who was using those tests improperly, and who was using them lawfully - so there was demonstrably no need to outlaw the practice altogether.
So what are necessary and proper means of enforcing laws? It's as I said before: having the proper executive authorities make investigations when they suspect a violation, having them level charges when their investigations lead them to believe that a violation has in fact occurred, and having the proper judicial authorities take it from there to its conclusion. Also, Congress, under the necessary-and-proper clause, can, for example, provide for the punishment for any acts of violence or fraud against the officers of the United States, since it would be improper (i.e. potential conflict-of-interests) for them to have to rely on the good graces of state governments to provide them with the proper security so they can do their jobs. Also, Congress would have the power to exempt them from various state laws and practices that might hamstring them (which I think McCulloch vs. Maryland was concerned with, or was it Maryland vs. Bank of the United States?), and to hold them liable only in federal court (not state courts) for any crimes they might commit in the course of their duties. Necessity and propriety does not involve giving Congress entirely new categories of power over society, just for the sake of exercising delegated powers.
You're raising one of the most important constitutional issues that this country has ever faced. When selecting a means by which to accomplish a constitutionally permissible end, is the Congress restricted to only those means which are indispensable to accomplishing that end or is the Congress entitled to select any means so long as the means selected is not prohibited by the constitution?
President Washington had been in office less than two years when he had to confront this issue regarding formation of a national bank. Secretary of State Thomas Jefferson argued that the Federal Government had no constitutional authority to create a bank, whereas Secretary of the Treasury Alexander Hamilton argued in favor of the plan.
Jefferson argued that the “necessary and proper” clause only authorizes Congress to pursue its express constitutional objectives with means that are indispensable to achieving the objective:
“If has been urged that a bank will give great facility or convenience in the collection of taxes, Suppose this were true: yet the Constitution allows only the means which are "necessary," not those which are merely "convenient" for effecting the enumerated powers. If such a latitude of construction be allowed to this phrase as to give any non-enumerated power, it will go to everyone, for there is not one which ingenuity may not torture into a convenience in some instance or other, to some one of so long a list of enumerated powers. It would swallow up all the delegated powers, and reduce the whole to one power, as before observed. Therefore it was that the Constitution restrained them to the necessary means, that is to say, to those means without which the grant of power would be nugatory.”
In response, Hamilton argued that the “necessary and proper” clause should be read less restrictively:
"In entering upon the argument, it ought to be premised that the objections of the Secretary of State and Attorney General are founded on a general denial of the authority of the United States to erect corporations. The latter, indeed, expressly admits, that if there be anything in the bill which is not warranted by the Constitution, it is the clause of incorporation.
Now it appears to the Secretary of the Treasury that this general principle is inherent in the very definition of government, and essential to every step of progress to be made by that of the United States, namely: That every power vested in a government is in its nature sovereign, and includes, by force of the term, a right to employ all the means requisite and fairly applicable to the attainment of the ends of such power, and which are not precluded by restrictions and exceptions specified in the Constitution, or not immoral, or not contrary to the essential ends of political society."
After considering these arguments, President Washington agreed with Hamilton and approved the national bank. In McCulloch v. Maryland (1819), Chief Justice Marshall sided with President Washington and Hamilton on this issue:
”We admit, as all must admit, that the powers of the Government are limited, and that its limits are not to be transcended. But we think the sound construction of the Constitution must allow to the national legislature that discretion with respect to the means by which the powers it confers are to be carried into execution which will enable that body to perform the high duties assigned to it in the manner most beneficial to the people. Let the end be legitimate, let it be within the scope of the Constitution, and all means which are appropriate, which are plainly adapted to that end, which are not prohibited, but consist with the letter and spirit of the Constitution, are Constitutional.”
Who do you think was right about all this?