Say a novel is set 100 years from now, when a confederation of nations establishes a unified military force. (Yes, I know, but this is fiction after all) Would a military academy, training astronaut fighter pilots for combat in interplanetary space, in a space station in geostationary orbit automatically raise a red flag as not being credible? Would the natural environment at that location be a disqualifying factor?
100 years is a long time in technology; we've gone from Goddard's home-made liquid-fuelled rocket to plasma drives in that time, so even a relatively conservative extrapolation of current technologies will give you lots to play with.
TL;DR: it'll be fine, more or less.
The main issue with putting people in geosynchronous orbit is radiation exposure. There are three main sources of this... the van Allen belts, the sun, and the whole of the rest of the universe (approximately). In all cases, your station, your fighters and your fragile meaty crew will all be subjected to a continuous slew of high energy particles. You'll need decent shielding, both physical and ideally electromagnetic. The technology for the latter is lacking today, but various things are in the pipeline and in a century's time will surely be standard equipment. Don't forget to keep an eye on the space weather forecast, and head for the shelters if a CME is in the offing!
The second issue is rocket technology. Chemical rockets will require continuous supplies from somewhere, so that's either a massive and continuous spacelaunch program or in-orbit fuel refineries fuelled by some extraterrestrial source like asteroids. Nuclear rockets are hazardous radiation sources and need fuel that will probably have to be brought up from earth. Fusion-powered rockets would be best of all, but that's a breakthrough technology and in any case will still need supplies of specialist fuel and reaction mass. Probably you'll have to handwave in asteroid mining, too.
The final issue is "why?". What's great about geosync, other than the rule of cool? Low orbit is safer and cheaper to get to, and lines of sight are much shorter making use of transatmospheric fighters less implausible. What are your fighters fighting out there? If you want to be further away from earth for safety reasons, I'd sit in lunar orbit instead, or maybe in a Lagrangian point.
Answer those three questions, which aren't super challenging (except maybe the last!) and you'll be fine.
The issue of whether spacefighters make the slightest bit of sense I'll leave for another question, but suffice to say I think that missiles, lasers and coilguns are better, easier and probably cheaper, though I appreciate they take the human element out of things so they dono't always make for good storytelling.
Lower A Cable!
If space elevators exist in your world it would become very easy to justify the station being at geostationary altitude. Since space elevators must be based from a geostationary point, it is almost natural for your station to exist here.
This would also give a credible explanation for how massive amounts of material and personnel are moved out of the gravity well. Unless you have some very efficient propulsion systems or are building spacecraft in space from materials in the solar system it will be hard to convincingly have a fleet based on using rockets to travel to and from Earth.
I guess "political considerations" could trump logistics but that seems unlikely for a military base when there are so many better options. Of course this is also assuming you create credible shelter against the radiation environment at that altitude (covered well in other answers). A space elevator requires you to be located in geostationary orbit and the political considerations could determine where along the equator its base lies.
Assuming this means said fighters will be operating and fighting in space, then the most economical way to put human pilots in a starfighter cockpit is to base a trainer squadron on a space station. Not necessarily at geo-sync but at least already-in orbit, as this would save the fuel and risks of ascending from the surface and re-entering every training flight. In that case the best solution would be to fly up a fresh class of junior pilots for a few months. But probably this would be the last part of flight training with the rest of it taking place in-atmosphere and in flight simulators.
Basing a squadron somewhere means you're also basing their maintenance department there. That means of parts, personnel, food, fuel, tools.. a squadron goes through a lot. Training squadrons especially as n00b pilots tend to be hard on the planes and the flight schedule is non-stop. That would be a busy space station, so if there's an isolation / psychological conditioning aspect of the training that should probably be done elsewhere. Hope that helps
Yeah! In a salvaged alien space ship!
If you have a Star-Wars type world where space aircraft carriers sally out and are defended by little space fighters manned by humans, it makes sense to have some safe base of operations from which to train your fighter pilots.
Rather than just repurpose the ISS for this, you should use a space aircraft carrier for your orbiting school. Best: it is a carrier thought to be from your opponent, discovered adrift and towed back to Earth for study. Study of this giant craft goes on, but there is a lot of useful space on board which various factions of the military have commandeered for their own purposes. One of these purposes is your space school.
Siting a school full of enthusiastic young people in an alien environment orbiting the earth will be great for the story. Is there already an anime about that? It practically writes itself. The students and teachers interact and give the human energy to the story. The presence of spooky scientists / other military factions / weird stuff from poorly explored areas of the ship will inject lots of energy.