Free admission: this is only my take on the convention of making something -punk. Your idea may vary – and that’s fine! Hopefully you find something useful here anyway.
There’s been a proliferation of things tagged with -punk as an affix over the last several years, some of which seem to treat it as a way to say “This thing is like the original thing, but new!” That’s not quite what the concept originally was, as it seemed to be more a method of taking the rebellious aspects of punk and weaving it into something else – cyberpunk was applying the concept of punk to the near-future dystopia some people saw emerging from the course of civilization, a what-if critiquing our rampant consumer/capitalist society. Steampunk was a rebellious neo-Victorian altered timeline where steam technology and mad science were the rule, where feminism had transformed all the tight-wound propriety into something else, a what-if model asking about feminism and education. The most recent example of this model that I’ve seen is the Heartland Trilogy by Chuck Wendig; it’s only semi-jokingly referred to as cornpunk, and it’s an interesting twist on the world that could be.
How, then, do you apply the concept of punk to a game system? Depending on how in-depth you want to get, it can be as simple as a superficial veneer across the familiar – something like how Eberron for D&D has elemental-driven trains and airship but is still, at the core, D&D high fantasy – or it can be a complete rewrite of what the game offers.
Presuming that you want something a bit more complex than the ‘slap a gear on it’ of many steampunk fans (Note: I am not saying thing is wrong, even though the more hardcore steampunk fans will throw a screaming tantrum over it. You’re a fan if you say you are, and people don’t get to gatekeep your fandom.) you should look at what you’re trying to modify and think about what parts of it could be used to critique the modern world – the ragged what-if question that your punk explores.
Take your question and filter your chosen game through it; push it to the extreme, because pushing it to the limit is the goal here. Cyberpunk isn’t a Walmart on every block and corporate drones in suburbia, it is monolithic megacorporations that supplanted governments, a dystopia of capitalist hell, everything pushed to the brink of collapse by consumer culture. Steampunk isn’t Victorian England with feminism, it’s Victorian attitudes shoved into a cannon with feminism gunpowder and loaded aboard a lightning-powered airship crewed by sky pirates. The Heartland trilogy is floating cities of the elites and the world wrapped in seas of corn.
Take D&D high fantasy and ask what extreme point you can render from it; a world where wizards openly wield their power could become a world where nuclear war is one angry spellcaster away, where everyone wants the threat of spellcasting armageddon stepped down but doesn’t want to be the first to do so. Arcane research programs and espionage teams of magic resistant thieves deployed in deniable operations. Mass-produced magic items outfit armies that ride in battle-golems in proxy wars. All the specters of modern warfare, filtered through the lens of high fantasy. And then the players – caught in the middle of a proxy war between two major powers, their entire world devastated by the battle and the fallout, who have to rise to become peacemakers or warlords, rescuing refugees or conquering the resulting chaos.
Take Eclipse Phase, and chase the specter that many people fear already, the idea of a superintelligent AI and a robot apocalypse – or be the robots among people who fear a robot apocalypse, harassed, put down, and tormented by the biochauvanists who dominate the society. Push it to the limit – faced with unending harassment, do they make the best of it, fight back, or try to make a break for it? What kind of brutal challenges sit between them and their goals?
Any setting can be rewritten to explore an extreme what-if situation. What if the religiously empowered Jovian Republic of Eclipse Phase declares a Crusade against the rest of the solar system? What kind of nightmares can erupt in a holy war across an entire star system? What happens when other faiths fight back? What happens when another copy of you, edited into a religious fanatic, starts rampaging in the same habitat you’re in? What happens when you meet your fork, and they insist you’re the edited one – and they seem to have evidence?
Punk is about exploring those extremes. Consider giving it a try in some short-term campaign, as filler between longer games; the intense push it brings isn’t good for long-term games in most cases, but it can provide ample thrills for a month or two, and be set aside before people get burnt out on it.