Well, believe it or not, there were some in those days (including President Jefferson) who believed that the Supreme Court had not been empowered by the Constitution to tell the Congress that the Congress had incorrectly interpreted the Constitution when it enacted legislation. They argued in substance that courts were supposed to just read and apply the laws of Congress and that courts were not supposed to make the laws. As a matter of fact, I still hear politicians and others making that argument.
But Chief Justice Marshall agreed with you and held, "It is emphatically the province and duty of the judicial department to say what the law [including the Constitution] is." Marbury v. Madison (1803). And to give you an idea of just how talented Marshall was, he declared this power of judicial review in a case in which (1) the result was in favor of the Jefferson administration and (2) the result brought the litigation to an irreversible end. So there was nothing that Jefferson or anyone else could do about it.
Here's a part of Article I, Section 8 that I hadn't before mentioned:
"The Congress shall have Power . . . To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof."
In McCulloch v. Maryland (1819), the Supreme Court had an opportunity to provide some meaning to that provision. See if you agree with Chief Justice Marshall's reasoning in this case:
"The government which has a right to do an act, and has the duty of performing that act, must, according to the dictates of reason, be allowed to select the means, and those who contend that it may not select any appropriate means, that one particular mode of effecting the object is excepted, take upon themselves the burden of establishing that exception."
If you agree with Chief Justice Marshall's reasoning in McCulloch, who gets to select the particular means by which the Fifteenth Amendment is to be enforced? And who would have the burden of establishing that the means selected (i.e., the suspension of literacy tests) is an improper means of enforcing the Fifteenth Amendment?
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.