Technically, the equator isn't in Middle-earth at all. Middle-earth is actually a continent, not the whole planet. The name of the planet as a whole is "Arda", although it is sometimes referred to simply as "Earth".
Arda is, in fact, our own dearly beloved home, planet earth. Middle-earth eventually became the continent we know as Europe (as it happens, Tolkien once wrote, in a letter, that the destruction of the One Ring took place about 6,000 years ago). Obviously, the equator is several thousand miles south of the southernmost tip of Europe, and since Europe used to be Middle-earth, the equator of Arda was far to the south of it as well.
The fine folks at the LotR wiki have produced a map showing how the old continent of Middle-earth probably would have aligned with modern-day Europe:
Moving on to where the equator would have been on Arda:
In Tolkien's works, the equator is referred to as the "Girdle of Arda" or the "Girdle of Earth". The only references to it, as far as I can recall, are in The Silmarillion. In this book, I have come across two mentions of the Girdle of Arda.
The first reference isn't very helpful; it merely says that the sun and moon followed the Girdle as they traversed the heavens. The other reference is only slightly more useful: it says:
Now Fëanor led the Noldor northward, because his first purpose was to follow Morgoth. Moreover Túna beneath Taniquetil was set nigh to the girdle of Arda, and there the Great Sea was immeasurably wide, whereas ever northward the sundering seas grew narrower, as the wasteland of Araman and the coasts of Middle-earth drew together.
- The Silmarillion, Chapter 9: "Of the Flight of the Noldor"
Túna is the hill housing the city in which the Elves lived in Valinor, and Taniquetil is the tallest mountain in Valinor. From this passage, we can see that the hill of Túna is near (nigh) to the equator of Arda (i.e., the earth).
This isn't especially helpful, however, because as far as I know, there are no canonical maps of Valinor, so there aren't any maps that show Valinor's position in relation to the position of Middle-earth. Therefore, we can't really say where the Girdle of Arda would have crossed the planet in relation to the area we are most familiar with (i.e., Middle-earth).
I'm not sure if this map is canonical, but it was drawn by Tolkien himself, so it must be close to canon status. It is known as "Ambarkanta Map V". The word "Taniquetil is visible on the far left, near the middle of the drawing, which gives us some idea of where the equator (or Girdle of Arda) would be; Middle-earth is far to the north of it.
We can, on the other hand, say that the Girdle of Arda did not cross Middle-earth itself, since we never hear anything about it; the climate of Middle-earth is also far too temperate for it to be anywhere near the equator. Furthermore, I don't have the quote handy right now, but Tolkien did say that The Shire is at roughly the same longitude and latitude as the British Isles in the real world. The canonical maps of Middle-earth focus on a relatively small area which is far to the north of the equator.
I would assume that the Girdle of Arda is somewhere to the distant south of Middle-earth, possibly in Harad.