What is an Anthroposophist?

One thing's for sure, it's not a person who says "I believe everything Rudolf Steiner said". That is the kind of Guru-worshiping complex he wanted to avoid. His desire would be that students think for themselves. 

To define an anthroposophist you'd have to define anthroposophy- it's not the kind of word a marketer would choose is it? 

There are many definitions of anthroposophy I've seen over the years- things like "Christian yoga after the coming of Michael", "the Wisdom in Man" or as it was termed during Steiner's theosophical days, "Rosicrucian Spiritual Science". But perhaps the best one is to be found in the Awakening to Community lectures.
In Dr. Steiner's own words:

The term "Anthroposophy" is really to be understood as synonymous with "Sophia", meaning the content of consciousness, the soul-attitude and experience that makes a man a fully fledged human being. The right interpretation of "Anthroposophy" is not "the wisdom of man" but rather "the consciousness of one's humanity". In other words, the reversing of the will, the experiencing of knowledge, and one's participation in the time's destiny, should all aim at giving the soul a certain direction of consciousness, a "Sophia".

Anthroposophy is a path of cognition from the spiritual in man to the Spiritual in the Cosmos:

Anthroposophy is a path of knowledge which would lead the spiritual in man to the spiritual in the Universe. It appears in men as a need of the heart and feeling. It must find its justification in that it can afford satisfaction to this need. Only he can acknowledge Anthroposophy who finds in it what he must seek out of his Gemut [a mind warmed by the heart].

Hence only those men can be called Anthroposophists who feel certain questions as to the nature of man and the world as life necessities in the same way as one feels hunger and thirst.

- Alfred Meebold's translation of Dr. Steiner's Leading Thoughts

Anthroposophy could only appear in the Consciousness or Spiritual Soul Age:

But such a coming together of human beings could take place only in our age, the age of the consciousness soul, and those who do not as yet rightly conceive the nature of the consciousness soul cannot understand this development..... It corresponds exactly to the developmental level of the consciousness soul period.

-Rudolf Steiner, Awakening to Community

"Anthroposophy is therefore the knowledge of the spiritual human being, or spirit-man, and that knowledge is not confined to man, but is a knowledge of everything which the spirit-man can perceive in the spiritual world, just as physical man observes physical things in the world. Because this second human being, the inner one, is the spiritual human being, the knowledge which he acquires may be called “Spiritual Science.” And this name is even less new than the name Anthroposophy. That is to say, it is not even unusual, and it would be a complete misunderstanding if anyone were to think that I, as has been said, or anyone closely connected with me, had coined the name “Spiritual Science.” The name is used everywhere where it is thought possible to attain knowledge which is not merely physical science, but knowledge of something spiritual. Numbers of our contemporaries call history a spiritual science, call sociology, political economy, aesthetics, and the philosophy of religion spiritual sciences. We use the name, only in a somewhat different sense, that is, in the sense that spirit is to us something real and actual, whereas most of those who nowadays speak of history, political economy, etc., as spiritual sciences, resolve the spirit into abstract ideas."

-Mission of Spiritual Science and of Its Building at Dornach, Switzerland