Academics treat philosophy like more of a discipline one does simply because one can. If you were to ask an academic what the purpose of philosophy is, they're likely to be confused by the question.
In contrast, the ancient Greeks, the Romans, and the Scholastics, believed that philosophy is an indispensable component of human life.
Thus after Scholasticism (1700 AD), and before today, something changed. But what? And when?
(Hat tip: Lucretius)
There's two sides to this issue. First, the Hellenistic period (and honestly, most Western philosophy up until the Enlightenment) was steeped in virtue ethics, and virtue ethics focuses on the development of proper (idealized) human character. Virtue ethics is inseparable from daily life, because daily life is the context in which virtue is both developed and expressed. When consequentialism and deontology entered the scene as competing theories, philosophical thought became more detached and abstract. Philosophers became less concerned with trying to 'live' the moment correctly, and instead sat back to mentally work out consequences, or devise systematic rules that could be applied across a range of situations. These more 'cerebral' schools took on prominence in academic settings, because academics settings naturally lean towards detached, abstract reflection. Note that there was a prominent philosophical rebellion against this — through Kierkegaard and Nietzsche into phenomenology, existentialism, and associated movements, where the focus is on the primacy of 'experience' — but all of these movements found a much better home in social and critical theory than in philosophy proper. Academic philosophers kept to a detached, analytical, almost 'clinical' perspective, while those who wanted to 'live' philosophy went into more active, socially-oriented fields.
Second, there's a distinction between 'knowing' and 'understanding' that cannot be understated. One can understand something well without much knowledge, and one can have a tremendous amount of knowledge without understanding anything at all, but in general the accumulation of knowledge (theories, facts, and information) becomes a tool for developing understanding (practical, living wisdom). Given the amount of raw theory and information in the modern world, it's good and useful to have people who focus on gathering and organizing knowledge, without expecting them to be practitioners as well. I mean, we expect a gynecologist to know a tremendous amount about human reproduction without expecting him or her to be constantly engaged in the act of human reproduction. Such acumen is useful to us when we reproduce; it doesn't need to be founded in the gynecologist's own life experience. Likewise, academic philosophers can tell us a lot of useful material without necessarily living it on their own. The mere act of presenting what we collectively know in a (comparatively) accessible form has value.
I will enlarge upon Nuclear Wang's reply by turning the question around a bit, as follows.
When did natural philosophers (also known as scientists) decide that philosophy was not useful, and why?
I believe this change in thinking about the natural world occurred along with the development of the scientific method, when its practitioners grasped the idea that simply because we might think or philosophize that the world should behave in a certain manner, based on logic or pure reason and our intuition (and hence not requiring experimental verification), it did not mean that the natural world will behave in that manner. There was when the split between science (as "natural philosophy") and what remained, which we might call "philosophy", first started.
This split has widened over the centuries as more and more parts of philosophy were discovered to have sufficiently strong scientific underpinnings that they stopped being issues for philosophers to debate and instead became topics of scientific research.
Philosophy became more limited in scope during the 19th century. In the time of the ancient Greeks, philosophy encompassed all forms of knowledge, but specialized subfields of physics, medicine, astronomy, and so on later emerged and became distinct from philosophy. In the past, philosophy encompassed pretty much all knowledge, but today, it's a much more limited field. I don't think anyone has ever declared that philosophy is "not useful", but it stands to reason that a limited subfield is less useful than a field that encompasses it. Algebra is not as useful as mathematics as a whole, classical mechanics is not as useful as physics, and botany is not as useful as biology. In this sense, modern philosophy only encompasses a small portion of classical philosophy, and could be argued to be not as useful. Philosophical questions have become more limited in scope, but not necessarily importance.
In the old joke, a college president complains to his physics professors that they are always asking for expensive equipment. All the mathematics teachers need is pencil, paper, a wastebasket, he says, and the philosophers don't even need the wastebasket.
There has always been a feeling among many that philosophy is useless or worse, and one can find this sentiment expressed not only by Aristophanes, for example, but by several interlocutors in the Platonic dialogues. In "Thesis on Feuerbach," Marx famously complained that heretofore philosophy has only described the world, when the point is to change it.
Philosophy, after all, does not put food on the table, win battles, or construct machines. As others here have noted, philosophy did engender many academic disciplines, from physics and psychology to sociology, computer science, and modern economics.
But once these fields became mathematically formalized for control and prediction, in other words "useful," they were no longer part of "philosophy." Of course, the very idea that "physics is more useful than philosophy" is itself a philosophical assertion, not something you can prove within physics.
There is something about philosophy that is inimical to practical applications. One could think of it like art as an end in itself not a means. Once the argument becomes a "means to an end" it is no longer philosophy. Science, for example, is means to the end of understanding causes for purposes of prediction. But it is engineering that turns this into "useful" things, not the science per se.
Since philosophy considers all assumptions to be fair game for critique, it cannot arrive at the sorts of axioms that make science "work" and its findings transformable into technologies. That conditional "suspension of falsification" in the "working theory" defines a "cause" but not the cause of the cause, the infinite regress of "because" that compels philosophy.
Since Socrates, philosophy has been a kind of via negativa revealing what we don't know. Yet, this is useful, many would argue! It is a method of revealing our false assumptions and vast complexes or shaky beliefs. As Mary Midgley notes, philosophy is everywhere and a bit like plumbing. You just don't notice it or think you need it... until the assumptions start to break down.
There are three ways to know something: by observation; by inductive reasoning; and by revealed wisdom from a divine entity. (Even this distinction requires relatively sophisticated philosophy, of course.)
The key point of philosophy is that a reasoned argument can proceed from some basis (which may be observations, revealed religious truth, or hypothetical what-if scenarios) to the consequences. The basis for argument may have errors, but the chain of reasoning can be made without error.
Initially, the key point was to work out the corners of religion. If your god or gods gave you rules, how do conflicting rules work? What happens in exceptional cases? If following the rules properly affects your treatment in the afterlife, this is important. This is the purpose which rabbis and Jesuits, amongst others, would value religion for. And also, importantly, you can find what things you can do without your god or gods caring about it. When your king may want to levy taxes, start wars or whatever, this sets what might be within the king's authority and what is set by religion.
The Greeks weren't too dogmatic about religion though, and that set the scene for observing the world and working out the consequences of those observations. Other people such as the Babylonians had rules about what had to happen at equinoxes or other similar events, requiring them to observe the world to work out how to best follow their religion. That's where natural philosophy comes in.
Religions being what they are, sometimes they conflict with observed reality. They then have two options: either acknowledge the conflict and work out how to deal with it (generally by regarding the text as parable instead of truth, such as the Catholic Church accepting evolution); or by believing that this observation is a test of faith and we cannot believe what we see (such as the evangelical view of evolution). Either way, it tells you those limits.
And moving forward, atheism and agnosticism are relatively modern concepts. From the idea that there may not be any god, humanist philosophy has given us morality from first principles without needing any religious text. This is now the basis for law in most of the world - and where is isn't yet, it's generally on the way. This makes it still highly valuable. Medical ethics is one very visible application of this.
So in short, no-one has decided philosophy isn't important - because it still is.
Obviously one can challenge the premise a bit, but I think there an (excessive in my view) humility at times in contemporary philosophy. (Though there are plenty of counterexamples!)
I don't know precisely when this became the case, but in Anglo-American philosophy I think it arises from - epistemic deference to the natural sciences and empirical methods generally, and a soft form of cultural relativism which makes people nervous about absolute ethical or aesthetic claims.
