In fact the problem ends up generalizing to giving in to sexual impulses outside the context of marriage. These arguments from a Greece that has not encountered Judaism, long before the birth of Christ, are going to sound very familiar to those who have ever heard what the Catholic Church has to say about the subject. Partly this is because the Church has adopted them from Plato, whose work provided a great deal of inspiration in the early (and Greek) part of the Church's formation.
The Athenian begins by suggesting that the business of encouraging sexual morality is both easy and nearly impossible. It's easy (not as easy as he claims, sadly) to get people to live alongside the fair without having sex with them because we can see that they do not so engage in sexuality with their fair brothers or sisters, or own children. Thus, as long as we can persuade everyone to adopt the same wholly negative view of having sex with non-spouses that we have for incest, a society should in principle be able to attain the end.
The near impossible part is, of course, convincing everyone to go along with that view.
So he begins by proposing first arguments, and then legislation to enforce the conclusions of his arguments. Now the arguments he uses are much more familiar to most readers from Plato's Symposium. They are given here in a much more straightforward form, without the drama of Alcibiades' attempted seduction of Socrates and Socrates gentle rebuff of the younger man's attempt. Here the Athenian just lays out the principles Socrates used in his defense:
Ath. The friendship which arises from contraries is horrible and coarse, and has often no tie of communion; but that which, arises from likeness is gentle, and has a tie of communion which lasts through life. As to the mixed sort which is made up of them both, there is, first of all, a in determining what he who is possessed by this third love desires; moreover, he is drawn different ways, and is in doubt between the two principles; the one exhorting him to enjoy the beauty of youth, and the other forbidding him. For the one is a lover of the body, and hungers after beauty, like ripe fruit, and would fain satisfy himself without any regard to the character of the beloved; the other holds the desire of the body to be a secondary matter, and looking rather than loving and with his soul desiring the soul of the other in a becoming manner, regards the satisfaction of the bodily love as wantonness; he reverences and respects temperance and courage and magnanimity and wisdom, and wishes to live chastely with the chaste object of his affection.
This is an argument that assumes an equality between men and women, notice, rather than (as we often hear) that they are contrary: if male and female were contrary, then love between even man and wife would fall under the scheme that is used to reject love between the young man and the older man, or between the rich and the poor. (In the laws on marriage, the rich are actually supposed to marry the poor by regulation; but because it unites their households and their wealth, this reinforces equality. The argument here is that a rich man loving a poor boy creates an inequality so steep as to make true and virtuous love impossible between them.) The importance of equality to friendship will be revisited in Aristotle's ethics; here, the implication is both that men and women have an equality, and also that man and wife should seek to love each other's souls more than bodies.
Of course there must be children; there's already been quite a bit of legislation proposed about making sure they get to that part also. Perhaps strangely, though, there is a differential in age there: women are to marry between 16-20, and men between 30-35. For whatever reason, this does not strike the Athenian as the same kind of inequality that worries him when he is rejecting the Greek system of man-boy love.
We also get the usual arguments from nature against homosexuality, so that Alcibiades' general impulse is to be "utterly" rejected and suppressed on pain of losing citizenship and all honor. (This same punishment is to attend anyone who has sex outside of marriage in any way, e.g. a man who had sex with a female servant would be punished just the same way as a partner in a man-boy relationship; presumably also unfaithful wives.) So you get the argument that nature plainly forbids 'spending your seed on unfertile soil,' i.e. other men, but also on fertile soil where a crop is not wanted.
The arguments from nature are generally rejected these days, following one of two quite different lines. The first (often citing Hume) is that 'you cannot get an ought from an is,' i.e., that you cannot reason from facts about what happens to be true about nature to what ought to be true. Thus, nature is rejected as a source of moral argument (for Kant, in favor of practical reason).
That approach has always struck me as wrong. Every moral principle is going to be founded at least partly on facts about reality. Murder is only wrong, if it is, because you can die; if it were not possible to kill a man, we would not need to have rules against doing it. Kant's practical reason would find no purchase on murder in a world in which death was impossible. Likewise these sexual rules are only going to exist to regulate a nature we happen to have. It's only important to regulate sexuality because we can have sex; it's only important to take care that children are born into families that can support them because children are a thing that can happen with sex; and they have natures that require nurture for survival; and stable families provide... etc. Not only can you get an ought from an is, what is happens to be the only place you can get your oughts.
(Furthermore, reason itself belongs in the realm of what is. So you can't escape the problem by flying to practical reason or even pure reason: you're still appealing to a part of what exists to obtain whatever oughts you wanted.)
The second approach addresses arguments like the Athenian's other point, which is that we ought to be better than beasts, and yet we can observe in birds both an avoidance of homosexuality and faithful mate-pairing. If the birds can manage to suppress their bestial lusts so well, why shouldn't human beings be able to do it?
The usual counterargument is that, you know, birds may do this or that; but humans have much closer relatives in nature, and they do seem to engage in homosexuality, non-paired sex, and a lot of other things. Why cherry pick the birds? If you're going to reason from nature, why not from the natures closer to our own?
The Athenian doesn't receive this challenge, but I'm not sure what kind of counter-argument he would raise against it. I think he feels like he's obviously proven the immorality of male-male sex already, both on natural grounds of the first type and also on the Symposium grounds that it leads to a lower type of love. But these arguments stand as a challenge to the first kind of natural arguments too; it may be that the social benefits of intercourse among members of a troop offer a second natural function for human (and near-human) sexuality, one that is proper to reason from as well as reproduction is, and according to its own facts. Here, for example, one might say that homosexuality has an advantage in that it can't produce offspring, so that the social goods (whatever they are) can be had without running the risk of a child who wouldn't be supported. If that kind of argument were right, Alcibiades' offer might be reasoned to be superior to engaging in sex with someone where reproduction would be possible. It's natural to beings of our sort in a way, apparently, and this form of it can attain that second end more efficiently and with fewer unwanted side effects. Thus, Alcibiades could argue that it was much better to pursue with him this second social good, instead of the first reproduction good; when the first good was wanted, well, that's when it would make more sense to prefer a female. Practical reason, right?
(The Athenian doesn't raise disease, as Catholics often used to do, as an additional natural argument against homosexuality. I don't know if they had the medical apparatus to appraise it; we are still close to Hippocrates, who was the first to suggest that disease occurred for natural causes rather than out of spiritual corruption, curses, etc.)
The Christians have an argument against that which Plato doesn't have, which comes out in Aquinas but is adapted from Augustine's thinking on the problem of evil. For Aquinas, there are not two but three natural goods to be had from sex. Yet to pursue only one or two of them, rather than all three together, is a privation of the fullness of good that God intended for the act. From Augustine, we learn that evil is in fact a privation of God's intended fullness of good; this explains why evil can exist in a world created by a perfectly good God. God gave you the tools to attain all the goods, but you chose to prefer only a subset, and thus we fell away from the perfection; and everything we call evil is an example of us attaining only some of the goods God enabled.
Thus, Aquinas can argue (but Plato cannot) that the existence of a second good doesn't imply that you can elect to pursue only one or the other; and since the goods must be pursued together, they can only be pursued at all in a case in which all are possible, e.g., in a heterosexual marriage in which all the goods can be obtained. That happens to be exactly what Plato's Athenian wants as well, but I think he lacks the philosophical apparatus to get there.
This approach ends up endangering the Symposium argument in another way: why should I pursue the spiritual and chaste soul-love of my beloved only, rather than pursuing that good and the erotic goods together? Aquinas can still point out that it is only in marriage that you and your beloved can do that without giving up the fullness of the goods on the table (to include the good of having a stable situation for any children produced, which is a very great good for them!). Yet it does seem as if the Symposium is suggesting a kind of privation by avoiding the erotic goods; maybe the answer lies in elevating the erotic to a level of equality, and always in the embrace of the love of the spiritual and virtuous good of your beloved as well.
(A contemporary reader might ask: what about birth control? It does not come up for the obvious reasons, but if you are seeking your own moral guidance from these arguments today, you'll have to decide what you think about that. The Church obviously teaches that it is definitely a privation because it disables the reproductive function that is one of the goods -- indeed, Aquinas calls it the principle end.)