A more recent perspective
Overall I think most modern justifications of astrology are nonsense. The outward motions of the planets do not cause changes in our lives through some influence like gravity or light. Trying to find a justification of this sort is futile and people should give up. It is like assuming that a that is about to strike 12 is causing one to be hungry, rather than that one's own internal body cycles are doing that, and that the clock is just an outward timekeeping device that gives us some indication of where we are in those cycles. The stars themselves don't compel or impel. Attempts to show otherwise will almost certainly never work. Consider that the exerted gravity, light, etc. from other people around you is probably greater than these things exerted by distant planets. Astrology is not a pseudoscience. People on both sides should stop pretending it is.
At the same time many criticisms of astrology are aimed at these nonsense defences. I remember one Sky and Telescope article suggesting that we could debunk astrology by suggesting that we might be able to make predictions by the placement of jets around busy airports, but debunking astrology by comparing it to a high-tech equivalent of augury strikes me as a very narrow rebuttal and certainly not one that would have been persuasive to Plato or Cicero (who took great pride in his auguries).
The reason here is that the ancient world saw astrology through a very different lens than we do today. Different fundamental assumptions about the nature of the world were at play and these assumptions are beyond proof in both systems of thought and while there are commonalities perhaps based on fundamental thinking processes, the differences are significant.
In every culture in every time, the human body and the universe have been seen as metaphorically equivalent. If the universe is a machine, so is the body. If the universe is a tree, so is the self. If the universe is a set of concentric spheres, so is the self. This fundamental, universal equivalent is important both in the acceptance of astrology in the ancient world and the rejection of it today. We see the universe and the human body as both impersonal machines, while Plato saw both as concentric spheres, seven (the planets in the outer world) of difference and one (the fixed stars) of sameness. It's fairly clear here also that what Plato intended to be an illustration of the human condition using astrology as a base was also at least one inspiration for Jung's idea of the collective unconscious (Jung makes repeated references to Plato's cosmology in his work, and this is virtually identical to Jung's idea of the a priori self along with the component which is universal based on being human).
Plato thus saw the human body and the universe as personal systems rather than impersonal ones, and here's the basis essentially for treating the motions of planets as indications of what is going on for the individual. Much as a clock might indicate it is lunch time, planets might indicate cycles within the self. This is similarly echoed by Paracelsus many centuries later. If these are the case, then jets around airports, or flights of birds are just as valid as astrology, and indeed flights of birds are a classic (and classical) method of divining prospects of success right along with astrology.
It is fundamentally impossible to prove whether the universe is personal or impersonal though the latter is a convenient assumption for the standpoint of modern science. We assume this because we find it useful, not because we can know it to be true. Moreover while deductive logic may be simple to formalize, inductive logic is not, and it is no less central to science and the like than deductive logic is.
Inductive logic arises, according to Heisenberg (see his book "Physics and Philosophy") and others, from the interplay of data and fundamental, unprovable assumptions. While you can disprove astrology from an enlightenment frame of reference, you cannot disprove the ancient Greek frame of reference, nor can you disprove astrology from that frame of reference. Astrology, augury, and a few other related disciplines necessarily follow from some frames of reference, namely those which see the outer world and the self as isomorphic (and hence see the body as fundamentally an aspect of the self).