According to the Jewish philosopher Philo of Alexandria, who was a contemporary of the Jewish historian Josephus, and who both lived in the First Century and who both commented extensively on the history of the Jews, the understanding was that Miriam's chorus of women had simultaneously complemented the chorus of men, and so Miriam responded to them, that is, as the leader of the chorus of women, she responded to them [the chorus of the men led by Moses] while they [the men] were still singing. Philo writes the following in regard to the passage in question.
For this mercy Moses very naturally honoured his Benefactor with hymns of gratitude. For having divided the host into two choruses, one of men and one of women, he himself became the leader of that of the men, and appointed his sister to be the chief of that of the women, that they might sing hymns to their father and Creator, joining in harmonies responsive to one another, by a combination of dispositions and melody, the former being eager to offer the same requital for the mercies which they had received, and the latter consisting of a symphony of the deep male with the high female voices, for the tones of men are deep and those of women are high; and when there is a perfect and harmonious combination of the two a most delightful and thoroughly harmonious melody is effected. And he persuaded all those myriads of men and women to be of one mind, and to sing in concert the same hymn at the same time in praise of those marvellous and mighty works which they had beheld, and which I have been just now relating. At which the prophet rejoicing, and seeing also the exceeding joy of his nation, and being himself too unable to contain his delight, began the song. And they who heard him being divided into two choruses, sang with him, taking the words which he uttered. (emphases added)
What is remarkable is that Philo's commentary was not addressing the variant readings or ambiguous understandings of this passage, which is what we are trying to do today. Instead, his flowing narrative addresses this portion of the Torah with glib commentary that the male and female choruses had sung one with the other in complementary unison.
Finally, while Targum Onkelos (its variant reading), the Septuagint, and even the Latin Vulgate read that Miriam answered them (feminine plural), the earliest extant record of this passage known to exist in the world comes the Dead Sea Scrolls fragment from 4Q14 (Plate 1074, Frag 1, B-295437), which indicates that the Miriam answered them (masculine plural). In other words, this earliest witness from the Dead Sea Scrolls lends credibility that Miriam responded to them [masculine plural].
Yonge, C. D. with Philo of Alexandria. (1855). The works of Philo (Vol III). London: Bohn, 129.