My anti-empiricism isn't fundamentally because the scientific method doesn't work. It's because it doesn't represent a sufficiently solid epistemological foundation. Let me try and explain.
Descartes, when he famously said, "Cogito ergo sum," was trying to derive a philosophical foundation from which it was possible to derive everything else in the universe - he was concerned about epistemology (how we can know). The statement is a good start, but unfortunately, you can't get past that foundation. You can't know that the rest of the universe is anything other than some aspect of your own consciousness.
The empiricist approach is to put this issue to one side, and say, "Well, the universe seems to behave itself - it seems to behave as though there is something really there, that acts in a tractable and reproducible way - I can do experiments on it, I can make observations in ways that I hadn't previously imagined, and it all hangs together properly."
The problem with this is, how can I know that this isn't simply my own consciousness constructing a world around me? How can I know that I am not a part of some Matrix-like construction? How can I know that, in fact, there isn't a god, and I am seeing exactly what he/she/it wants me to see? And the answer to all of those questions is, you can't. You start from your own consciousness, and make a leap of faith to conclude that there is a real universe beyond you. You then make another leap of faith to conclude that the real universe is all that there is - and in many ways, this is the oddest one. Given that you had to make a leap of faith to conclude that there was a universe at all, how can you then have any confidence that there is nothing more than the universe?
So what's the alternative? What do I believe that isn't empiricism?
Start instead from the proposition that there is an external absolute, a prime mover (if you like). Yes, this is a faith position, but so is the belief that there is a universe there at all, and nothing else. Both are presuppositions. Both are foundations for knowledge upon which the rest of our understanding of the universe is built.
Francis Schaeffer, in an essay entitled "Is Propositional Revelation Nonsense?" makes the case for the reasonableness of this position, and I could do no better than he has already. In addition to which, I need to go to work now ....