I don't have it available to me (and besides, I'm a Norton boy, but I'm not on my own continent right now so I don't even have that text on me), but here's what Grove Online has to say:

In the Middle Ages the term derived from and was synonymous with ANTIPHON. After the Reformation the term denotes a polyphonic setting of a sacred English text, normally sung by the choir after the collects at Matins and Evensong; the text is freely chosen, most often from the Bible (especially the psalms) or from the Book of Common Prayer. The connection between Latin antiphon sung within the Office and English anthem sung as an appendage to Matins or Evensong is found in the Commemoration, Memoiral or Suffrage in which the antiphon was the most important musical element. In the medieval liturgy a Commemoration, Memorial or Suffrage was often appended to the main Office (e.g. Lauds or Vespers); this observance normally consisted of Benedictus or Magnificat antiphon, versicle and response, and collect – effectively a truncated Office commemorating an intention additional to the main Office (e.g. the saints, the dead, or a saint remembered on that day but not taking precedence in the main Office; see LU, 260–61, 273–7 for Roman Catholic equivalents).

I am fairly confident that the anthem was never a responsorial exercise. The early anthems by the big guys (Bird, Tallis, Tye, Batten) certainly seem strange when thought of as being performed in any sort of a responsorial style. I guess you could say that verse anthem are responsorial in a way (considering This is the Record of John and others in that vein, I suppose that an accompanied solo voice and choir are responsorial in a loose sense), but that could be contested. For the sake of argument, we'll call the verse anthem responsorial, but I still don't think that its responsoriality comes from the practice of singing antiphon in any real sense, particularly considering that Grove says of the verse anthem:

Easily the most significant development of the period [1565-1644, twenty years or so on from the earliest English anthems] was the creation of the ‘verse’ style, in which verses for solo voices and instrumental accompaniment (normally organ) alternated with passages for full choir.