Federalist Paper #75 deals with this objection directly. To wit:
Another source of objection is derived from the small number of persons by whom a treaty may be made. Of those who espouse this objection, a part are of opinion that the House of Representatives ought to have been associated in the business, while another part seem to think that nothing more was necessary than to have substituted two thirds of ALL the members of the Senate, to two thirds of the members PRESENT.
In this paper, Hamilton first question whether the Executive or the Legislative branch should deal with this. He writes:
The power of making treaties is, plainly, neither the one nor the other. It relates neither to the execution of the subsisting laws, nor to the enaction of new ones; and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations, which have the force of law, but derive it from the obligations of good faith. ... The qualities elsewhere detailed as indispensable in the management of foreign negotiations, point out the Executive as the most fit agent in those transactions; while the vast importance of the trust, and the operation of treaties as laws, plead strongly for the participation of the whole or a portion of the legislative body in the office of making them.
His compromise was thus to have both branches play a role. The executive needs to be the decisive actor, but the legislature would temper him.
Leaving it to just one man would be a King-making offense, not safe enough to leave to one man, or even just to the Senate would have left us open for tyranny:
To have intrusted the power of making treaties to the Senate alone, would have been to relinquish the benefits of the constitutional agency of the President in the conduct of foreign negotiations. It is true that the Senate would, in that case, have the option of employing him in this capacity, but they would also have the option of letting it alone, and pique or cabal might induce the latter rather than the former. ... hough it would be imprudent to confide in him solely so important a trust, yet it cannot be doubted that his participation would materially add to the safety of the society. It must indeed be clear to a demonstration that the joint possession of the power in question, by the President and Senate, would afford a greater prospect of security, than the separate possession of it by either of them. And whoever has maturely weighed the circumstances which must concur in the appointment of a President, will be satisfied that the office will always bid fair to be filled by men of such characters as to render their concurrence in the formation of treaties peculiarly desirable, as well on the score of wisdom, as on that of integrity.
And this brings us to the crux of the matter. Simply put, the founders didn't trust the House as much as the Senate.
The remarks made in a former number, which have been alluded to in another part of this paper, will apply with conclusive force against the admission of the House of Representatives to a share in the formation of treaties. The fluctuating and, taking its future increase into the account, the multitudinous composition of that body, forbid us to expect in it those qualities which are essential to the proper execution of such a trust:
- Accurate and comprehensive knowledge of foreign politics;
- a steady and systematic adherence to the same views;
- a nice and uniform sensibility to national character;
- decision, SECRECY, and despatch,
are incompatible with the genius of a body so variable and so numerous. The very complication of the business, by introducing a necessity of the concurrence of so many different bodies, would of itself afford a solid objection. The greater frequency of the calls upon the House of Representatives, and the greater length of time which it would often be necessary to keep them together when convened, to obtain their sanction in the progressive stages of a treaty, would be a source of so great inconvenience and expense as alone ought to condemn the project.
Put simply, the Senate, being drawn from the legislatures of all the states, and not subject to something as vulgar (in the sense of common as opposed to profane) as the House, was better able to balance the sophistication of statecraft with the need for multiple voices.
In other words, the Upper House (the Senate) had the chops - and the House was simple a bunch of bumpkins who couldn't hack it.
 For those unaware, the Federalist Papers were a series of essays written by prominent "Founding Fathers" written to help secure passage of the Constitution. They were written within weeks of the first draft of the Constitution have been sent to the States, and before it was ratified. They were published with the explicit reason of securing passage and explaining this new "Constitution" to America. The Supreme Court of the United States often looks to these papers when an Article is in question. Because of their timeframe, however, nothing in the Bill of Rights or any of the other subsequent amendments is handled in them.