I found a paper on this subject on the Social Science Research Network. This paper discusses what is known of Gödel's supposed Constitutional loophole, and proposes and discusses a number of hypotheses as to what that loophole might be.
The first thing to note is the lack of concrete evidence as to what this loophole is. Though several accounts indicate that Gödel believed he found a loophole, none seem to contain any details as to what that loophole was, none were transcribed around the time of his study of Constitutional Law, and many seem to disagree on minor details. Nevertheless, we can examine the proposed loopholes, and determine if any of them are worth our concern.
The author of the paper proposes that Gödel believed that the Constitution permitted dictatorship through amendment; specifically, that by amending the rules of amendment in the Constitution, Congress and the states could make future amendments trivially easy to implement. This could allow the US to become a dictatorship (or otherwise transform its government) very easily.
This "loophole", it seems to me at least, is obvious, and not particularly threatening. Though it does mean that Congress and the States, through supermajority, could push the United States in the direction of a dictatorship, the ability to change our form of government is the plain purpose of Article V. There is no reason to think that the Constitution would only allow good amendments; indeed, many of the amendments that existed at the time of Gödel's citizenship test (in particular, the 2nd and 11th Amendments) are highly controversial.
Furthermore, as the article suggests, such problems may not even be theoretically solvable. Any clause that would prohibit the amendment of the rules of amendment could itself be amended. And ultimately, if enough people want to change their form of government, they could simply ignore the Constitution and establish a new one. This was, after all, what happened with the Articles of Confederation. Changes to the Constitution requiring approximately this level of coordination hardly count as "loopholes".
The author then proceeds to discuss more mundane controversies of Constitutional law, and dismisses the notion that these were the kinds of controversies of which Gödel was thinking. Indeed, these potential "problems" hardly count as loopholes, especially when it is often vehemently argued that these "problems" are the exact outcomes the Constitution intended.
In particular, the author dismisses the notions that Gödel may have been concerned with a broad interpretation of Congress' power under the interstate Commerce Clause, a broad interpretation of the President's power under the Executive Vesting Clause, and a broad interpretation of the Judiciary's power under Article III.
Without going into comprehensive review of the vast amounts of case law on these issues, it is fair to say that these can't be considered loopholes. Though there is controversy about the power of each of the branches of government, the argument generally goes that either the Constitution doesn't grant the government broad powers (see Hamer v. Dagenhart, Youngstown Sheet & Tube Co. v. Sawyer, and Lujan v. Defenders of Wildlife for narrow interpretations of the legislative, executive, and judicial power, respectively), or that it does and that this is the intention of the Constitution (Wickard v. Fillburn, Korematsu v. United States, Cooper v. Aaron).
In either case, there is no loophole. Either the Constitution prohibits a power, or intends it; it doesn't allow it unintentionally.
Finally, the author proposes that Gödel may have been concerned with Congress' power to admit new states. As this only requires a Congressional majority, a Congress seeking to pass a Constitutional amendment without the consent of the states (Article V requires that 3/4 of the states ratify an Amendment) could, by simple majority, create 150 new states, potentially containing as their population only members of Congress favoring the new Amendments. These puppet states would then presumably all agree to ratify any Amendment.
However, this would be difficult, if not impossible. First, it would require the consent of some of the states, which is exactly the kind of thing that Congress, in this scenario, would be trying to avoid. Article IV, Section 3 of the Constitution says:
no new states shall be formed or erected within the jurisdiction of any other state; nor any state be formed by the junction of two or more states, or parts of states, without the consent of the legislatures of the states concerned as well as of the Congress.
Note the emphasized text; the creation of new states would require the consent of the states in which they were created. Furthermore, even they could find a way to construct 150 new states with the consent of fewer than 3/4s of the states, the creation of 150 new states is not a trivial task, requiring the approval of a majority of Congress each time.
And finally, even if Congress successfully managed to create an appropriate number of puppet states, a supermajority would still be required to amend the Constitution; in the end, Congress would have made the amendment process much more difficult, not less difficult.
Ultimately, there is no record of what Gödel may have found or thought he found in the Constitution, and examination of potential hypotheses as to the loophole has not generated any cause for concern. The American Constitution is not a long document, and the entire legal field has been dedicated to interpreting it and pushing its limits for well over two centuries. If there were a fatal flaw, we likely would have found it by now.