Proton-proton fusion happens at energies around 15 keV. The LHC currently operates at an energy of 13 TeV, which is literally one billion times larger. Fusion is one of the lowest-energy processes that could occur at the LHC, and most of the interesting reactions that are studied there go way beyond nuclear fusion.
A proton is made up of fundamental, pointlike bits of matter called quarks, held together by force carriers called gluons. These quarks are bound together with a certain binding energy; the quarks are also relatively light, so this binding energy actually makes up most of the mass of the proton. In other words, the binding energy of the proton is roughly equal to the mass of the proton.
When you collide protons at an energy that's far below its mass, the proton acts as a single object. This is the regime in which nuclear physics typically occurs; all of the reactions that break nuclei apart, and the energy that holds them together, are typically small enough that it works to treat the proton as a single unit most of the time. This includes nuclear fusion; the mass of the proton is 938 MeV, and the amount of energy required for nuclear fusion (15 keV, as we said) is tens of thousands of times smaller than that.
If we instead decide to collide protons at energies that are higher than a few times the proton mass, then there is now enough energy in the reaction for the quarks inside the protons to interact with each other directly. We can no longer treat protons as single objects, since the energy involved in the reaction is enough to expose its inner components. As the collision energy becomes higher and higher, the binding energy of the proton becomes less and less relevant, and the picture of two dense clouds of quarks and gluons interacting becomes more and more accurate. These reactions have enough energy to produce all manner of exotic matter, particles that you could never find in a nucleus, nor could you create them by nuclear fusion. The current energy of the LHC is roughly ten thousand times greater than the mass of the proton; at those energies, the proton's mass and binding energy are completely negligible, and the interesting reactions are entirely driven by individual quarks and gluons interacting with each other. The proton is merely a vehicle for us to deliver bits of fundamental matter to the collision site. If there were a way to just collide free quarks and free gluons, without all the mess generated by the proton, many particle physicists would jump at the chance. (Unfortunately, this turns out to be impossible.)
There's certainly lots of extremely interesting physics to do at the nuclear-physics scale, and our understanding of the dynamics of nuclei is the subject of a great deal of active research in other experiments. But that's simply not what the proton-proton collisions at the LHC are meant to explore. The proton-proton collisions are actually better thought of as quark-quark collisions, or quark-gluon collisions, or gluon-gluon collisions. They're not meant to study protons, which is why the protons are often totally destroyed in the collision, transformed into exotic matter which then decays back into ordinary matter. Proton-proton collisions are meant to make conclusions about the interactions of fundamental particles, and that requires acceleration to extremely high energies, far above what you would need, or want, for nuclear fusion.
So, given this, why aren't LHC physicists worried about triggering nuclear fusion? The answer is fairly straightforward: though each individual proton has an energy a billion times larger than the fusion threshold, the total amount of energy that is released into the surrounding area is still rather manageable, on a macroscopic scale. After all, 13 TeV is still only about a microjoule of energy, which is around a billion times less than the amount of energy that the Sun imparts to a square meter of Earth every second. That said, there are around 600 million collisions per second happening, so you definitely don't want to be standing anywhere near the interaction point. This is especially true since the individual particles of radiation released are much higher in average energy, meaning they're much nastier in terms of damage to life and inanimate objects than the radiation from the Sun. Because of this, the detector electronics have to be specially designed to deal with this extreme high-radiation environment; human access to the experimental hardware is also very tightly controlled, and is completely forbidden when the accelerator is running. But ultimately we're talking about around a few kilowatts, at most, of radiation released into the environment at each collision site. That's a pretty human-sized amount of power, and is roughly equivalent to the heating power of a large space heater (but, again, in a much more damaging form than the heat released by a space heater). This was by design - the collision rate at the LHC was chosen partly so that it would be feasible to build a detector that could withstand the influx of radiation. Nuclear explosions require many, many reactions all occurring at once, which is why they have such destructive power. The LHC collides at most a few individual protons at a time.
So, given all this, the LHC would make a terrible fusion reactor. Its energy is much too high to actually reliably trigger nuclear fusion, and the energy released in collisions is miniscule compared to the energy required to keep the beams running, so it would be incredibly inefficient.