Some astrological predictions prejudice the person concerned or give him extra mental strength and that becomes a crucial factor in the prediction coming true. In other words, sometimes astrologer ends up playing a critical role in an astrological prediction/indication coming true. This kind of cases can hardly be used to support astrology's cause.
However, there are cases too, when astrologer has absolutely no role to play in a prediction coming true. For example, I predicted Indian cricketer Sachin Tendulkar's performances for Oct 1995-Mar 1996 with a 100% accuracy on USENET. Before India's cricket series in England in the summer of 1996, I predicted on USENET that Sachin Tendulkar would do well on the third day of the first test and he hit a century on that day!
You may dismiss these predictions as hit-or-miss random guesses that happened to come true. That's a valid point of view. After all, the probability of Sachin Tendulkar making a test century on a given day is probably of the order of 1/15 (assuming he makes a century once in 3 matches). This probability is not low enough to make to my prediction an "amazing" one. It might've come true "by chance"
But there were several cases in which the non-astrological probability of my prediction, which eventually came true, was of the order of 0.0001-0.001!!
If, by looking at someone's chart, I figure out that he had serious digestive troubles during a month, what's the probability of it coming true if it were a random guess? Assuming that 10% people experience serious digestive troubles in life and that serious trouble comes to those for one month in a span of 3 years, this probability is 1/(3*12)*10/100=0.0028 (i.e., 0.28%).
Let me give an example from my formative years, that I am very fond of.
I once predicted three months ahead that an event important in a close friend's professional and financial life might happen during a three-day period. At that time, he was working in India and, being very satisfied with his job, he wasn't looking for a job change. He didn't even have a typed resume then! However, on the first day of the three-day period, he got a phone call from a consultant (who heard about him from someone) who had an overseas client interested in his background! Assuming that one out of thirty get an unexpected call like this and that it happens once in 3 years, the probability of a random prediction like this coming true is 1/30*3/(3*365)=0.00009 (i.e., 0.009%).
By looking at his horoscope, I asked a friend if his elder brother had a child in late 1989 or early 1990. He told me his elder brother indeed had a child in Feb 1990. I asked him if his elder brother had some failure in love, love problems and resultant emotional and health problems during late 1984 and early 1985. He told me his elder brother's girlfriend whom he had been dating for 4 years died in Jan 1985 and his brother was highly depressed. These predictions, it would seem reasonable to believe, are too good to be a mere fluke.
In my short career in astrology, I made quite a few successful predictions that had a very low non-astrological probability (of the order of 0.00001 to 0.1). While this need not prove to rational minds that astrology is true, it should atleast create doubts that astrology is perhaps not completely a superstition.
Personally, I have seen many a proof that astrology has some worth in it.