Swift is a master of satire...and ethos.
He begin by establishing himself as a man who has spent years studying the problem, and offering evidence that he has "maturely weighed" the various schemes of others who have offered solutions, and found them "grossly mistaken in the computation."
His plan also proposes to "prevent those voluntary abortions" so many of these poor women have, and he points out that we can all agree that murder of one's own child is horrible. Thus, he notes and agrees with the values of his audience, establishing further ethos. Along these lines, he points out that the children of the poor quickly become a scourge on society, being forced to turn to begging and stealing themselves, which is another thing no one wishes.
At the end of the essay, he points out that he has nothing to gain himself, clarifying that his proposal is for the good of the people of Ireland only:
I have not the least personal interest in endeavoring to promote this necessary work, having no other motive than the public good of my country, by advancing our trade, providing for infants, relieving the poor, and giving some pleasure to the rich. I have no children by which I can propose to get a single penny; the youngest being nine years old, and my wife past child-bearing.