So when you say "the Zoroastrians", I suppose you mean the Parsi communities. And the truth is that nobody really knows exactly when. There's no doubt that Zoroastrianism got more ethnocentric throughout antiquity, especially once its rivalry with Christianity got significant. But there's no prohibition against conversion laid out explicitly in any ancient text.

The ostensible explanation for some Parsi traditions is given in the Qissah-i Sanjan (composed in Mughal Court Persian in 1599 at the court of Akbar, with a probable partial precedent in the so-called Sanskrit Slokas, sixteen short poems that seem to have been composed by a Brahmin with a fairly superficial understanding of Zoroastrianism). The work makes their point rather aggressively: Zoroastirians sought refuge with the wise king Radi Jana:

In that region was a virtuous Raja who had opened his heart (lit. head) to holiness. His name was Jádi Rana; he was liberal, sagacious, and wise. A Dastur renowned for learning and prudence went to him with gifts and invoked blessings upon him and said: "O Raja of Rajas, give us a place in this city: we are strangers seeking protection who have arrived in thy town and place of residence. We have come here only for the sake of our Religion, for we heard that there was in this place a Raja descended from the beneficent Shillahras, ever renowned throughout Hindustan, who gave people shelter in his town and kingdom and regarded them with the eye of compassion. We were cheered by these tidings (lit. thoughts) and have approached thee under favourable auspices. We have now reached thy city in the hope of escaping from the Miscreants."

According to the Qissah, the Parsis were from Khorasan, famous for being the last Zoroastrian holdout state until the 9th century. It's possible that some Zoroastrians fled to Parsi communities after the Muslim conquest of the Karen realms of Khorasan, but the communities seem to have originated with Persian merchants, rather than refugees. The Qissah probably reflects tensions both at the time of its composition (when the famously harsh Shah Abbas the Great reigned in Iran) and in preceding centuries, with the Parsis presented as a natural, integrated part of India. Anyway, the story offers a rationale for certain characteristics of the Parsi:

What are the customs of your Creed, which of them are open and which concealed? Let me first of all see what your beliefs are and we will then arrange for your residence here. Secondly, if we give you shelter, you must abandon the language of your country, disuse (lit. cast aside) the tongue of Iran and adopt the speech of the realm of Hind. Thirdly, as to the dress of your women, they should wear garments like those of our females. Fourthly, you must put off all your arms and simitars and cease to wear them anywhere. Fifthly, when your children are wedded, the marriage knot must be tied at evening time. If you first give a solemn promise to observe all this, you will be given places and abodes in my city." When the Dastur heard all this from the Raja, he could not help agreeing to all his demands.

Although the story is occaisonally referred to in the context of conversion, it doesn't actually explicitly discuss it (but some who take it at face value infer that perhaps the Parsis agreed not to engage in proselytization in exchange for being offered refuge). Meanwhile, if we look at Rivayats from around the same time (essentially, legal advice sent by Zoroastrian clerics in Iran) we see that conversion was a thing among Iranian Zoroastrians:

Q: Can a grave-digger, a corpse-burner and a darvand (one of a foreign faith) become Behdins (i.e., be converted to the Mazdayasnian religion)?

A: If they observe the rules of religion steadfastly and (keep) connection with the religion, and if no harm comes on the Behdins (thereby), it is proper and allowable.

The Encyclopaedia Iranica offers the following reasoning for the now near-absolute opposition to conversion in Parsi communities:

Whereas the Persian Zoroastrian communities never explicitly opposed the acceptance of converts, since the 18th century Indian Parsi councils have generally refused to accept as Zoroastrians persons other than children of Parsi parents, though there have been sporadic rulings allowing acceptance of the children of mixed marriages. This rigor may be ascribed to the pressures of the caste structure in India, reinforced by the growing prosperity of the Parsi social-welfare system, a possible incentive for seeking admission to the Zoroastrian fold.

In Persia the majority of Zoroastrians lived in ex­treme poverty and suffered intermittent persecution up to the beginning of the 20th century. The question of conversion to Zoroastrianism would scarcely have arisen there.

This socio-economic explanation certainly seems like the most plausible one to me. The Parsis were among the favourite subjects of later British rulers, which seems to have instilled into the leadership of their communities not just an aversion toward conversion, but also a fixation on a notion of "racial" or "blood purity" that unfortunately is the primary reason for the very rapid decline of the community today.