The Greek text was first printed in Augsburg in 1610 by David Hoeschel, who was, incidentally, a student of Hieronymus Wolf (the guy who invented the term "Byzantine"). Hoeschel got the text from a manuscript he bought in Munich, so the text was already "available" in the sense that it could be read in manuscript form, although there are only a few medieval manuscripts of the text so it doesn't look like it was a hugely popular work. Hoeschel also translated the text into Latin, since most people in western Europe couldn't read Greek.

Another early edition was printed by Pierre Poussines in Paris in 1651. The Paris version was reprinted in the Patrologia Graeca by Jacques-Paul Migne in 1864, also with a Latin translation (like Hoeschel, Poussine translated it into Latin as well). The PG, along with its counterpart the Patrologia Latina, were hugely popular and important for scholars. Any big enough academic library probabaly has several full shelves just for the PG and PL!

So it was available in printed form in the early 17th century, and by the mid-19th century it would have been known to anyone familiar with the PG. A critical edition (i.e. comparing all the manuscripts to find the best version of the text) was printed in the Corpus Scriptorum Byzantinorum in Bonn (two parts, 1839 and 1878, also with a Latin translation). The most recent critical edition is in the Corpus Fontium Historiae Byzantinae (2001).

In the English-speaking world, the Alexiad was translated by Elizabeth A.R. Dawes in 1929, and then a more recent and better translation was made by E.R.A. Sewter in 1969. Sewter's translation is the popular Penguin version, which you can usually find in any bookstore. This version was re-edited and updated by Peter Frankopan in 2009.