For those of you who are unaware of the mind-body problem, it is basically trying to determine how the mind connects with the body, and can lead on to 'what is the mind' and whether it is a physical object (note - this refers to the mind not the brain). As in Star Trek, one can use the transporters to transport a person from one position to another, and the person is perfectly intact, this implies that Star Trek solves the mind-body problem. Is this the case and, if so, what is the solution to it?
Star Trek embraces whatever philosophy is required to tell the current story.
In the TOS episode "The Lights of Zetar" Enterprise encounters extant members of a dead civilization who existed for millennia as acorporeal minds searching for a suitable body in which to live out the remainder of their corporeal lives. This is an example of Cartesian or substance dualism.
In the TNG episode "The Schizoid Man" cyberneticist Ira Graves flees his dying body and causes his mind to run on a new substrate, the positronic brain and body of Lt. Cmdr. Data. This is an example of monism, in particular, physicalism.
On Star Trek: Deep Space Nine, we learn on the existence of the Prophets, acorporeal entities that exist at least partly outside time itself, though they do inhibit material bodies from time to time. This is (probably) an example of idealistic monism.
In the TOS episode "Turnabout Intruder", Janice Lester swaps ids with Captain Kirk, inhabiting his body while he inhabited hers. Substance dualism again.
In the TOS episode "What Are Little Girls Made Of?" Roger Corby transfers his mind to run on a new substrate: an android body. Physicalism again.
In the movie The Search for Spock we learn that Spock's katra, whatever that is, was deposited into Dr. McCoy to be later extracted by some mystical process. This is an example of property dualism.
If there's a mind-body philosophy that Star Trek has not yet embraced, give them time.
Star Trek's transporters use "Heisenberg Compensators" to circumvent the Heisenberg Uncertainty Principle that should prevent such devices from working (according to our current understanding of physics). It is quite likely that those same compensators account for the mind-body divide, though this is merely speculation on my part. One would imagine a culture that deals with telepathy on a regular basis would have far greater understanding of the Mind-Body Problem than our current society; if so, however, this has never been explored by the series in any medium.
The book "Is Data Human? The Metaphysics of Star Trek" goes into vaguely similar problems involving the transporters and issues of whether the person after the transport is the same person as the person before the transport, i.e. if who comes out of the transport is just a near-perfect copy of the person went in.
All I think we can gather from the existence of working transporters is that a perfect replication of a person's physical form results in their "mind" being identical. Whether that's because the mind exists physically within the body (which I don't personally believe) or the mind is an emergent property of the brain (which I do personally believe) isn't really answered either way.
I think overall "Star Trek" is pretty dualistic. People all have energy based souls, though they don't use the word. Those souls can be pulled out of people, and hang free as energy, or jump to other bodies...they aren't tied to the function of a particular piece of meat. In theory, transporters should be able to make copies of people, but they don't do this, and no matter the technobabble explanation, the overarching universe rule is that transporters shouldn't be able to make souls. And with the exception of Thomas Riker, they don't.
People with telekinesis get their abilities through structures in their brains (IIRC!). It stands to reason that the mind-body problem is also solved through such structures.
It is my understanding that spirits live in another "dimension" in Star Trek. A world in which spiritual energy and such exist more or less freely. There is the episode with body-controlling wisps, and the series of episodes about Wesley and the traveller that seem to support such a theory. This is however describing the mind.
There is no explicit mention of what ties such a mind to a body, but by extension of the telekinects being a property of the brain we can say that the brain has structures such that it ties the mind to the body. This is the most realistic explanation that still allows everything canon to be as it is.
Note: the Riker accident tells us that the mind is (in spacial..space) located where the body is, for the reflection and duplication to take place as it did. Alternatively the accident caused quite a stir in the world of spirits (like Q) and they replicated his spirit to cause the humans not to stumble upon such outlandish principles. (Q other extraspacial beings replicated Riker's soul to prevent confusion among humans for whatever purpose, possibly improved observation like Q's species likes to do)
It can take some mental gymnastics to work out solutions for nearly every episode. A flexible single approach as given above helps you do it :)