Airships, airships, AIRSHIPS
Airships are the aircraft most associated with this trope, so any analysis of it from a realistic perspective will have to start with them. For starters, airships are relatively slow-moving and can hover, so they can theoretically be boarded in midair, which helps with the whole The Sky Is an Ocean angle that a lot of these stories take. They also already use lots of nautical terminology in real life, especially in the case of the larger Zeppelins.Of course, in real life the greater speed of airships compared to trains and sea ships could not compensate for their lower carrying capacity and high costs. However there are a variety of ways that a setting could be made where airships are the primary means of shipping cargo and passengers, making the idea of aerial piracy a logical one.
Crimson Skies presents an alternate 1930s where the USA has balkanized into over 20 countries. The new borders severed roads and rails, leaving air the only means of commerce between states that don't share borders. Desert
Air is probably the best means of transiting desert that lacks the infrastructure for roads or rails. Though depending on the size of the desert, and whether there's any hostile natives, or Sand Worms, trains might be more cost-effective in the long run. If raiders are used as a reason for lack of infrastructure it's almost inevitable that they'll get their hands on some aircraft eventually. The flip side of this, of course, is that given the wide open spaces of the desert, airplanes might end up being more practical than airships considering there's enough space to make runways. Post-Apocalyptic
Most of the Earth's surface is irradiated. It's dangerous to travel between settlements on the ground and there's no room to spare for an airfield large enough for airplanes, fortunately zeppelins require much less room to take off or land. And there would be a lot of desperate people out there as well.Self-Explanatory. Venus
The atmosphere of Venus is so dense that a balloon filled with breathable air would float several miles off the lava-covered ground. In fact, there have been proposals
to build colonies in balloons over Venus. In addition air-breathing engines wouldn't work in the CO2-heavy atmosphere, leaving aircraft with electric props (or gliders) and severely limiting the range of planes.Heavier-Than-Air Options
If, on the other hand, airships don't exist in the setting of your story, or it takes place in something closer to the present day rather than an alternate history or the past, there are still some ways this trope can be pulled off with heavier-than-air aircraft (which is to say, airplanes and helicopters). These portrayals often lean more towards the Ruthless Modern Pirates side of things by default, since airplanes typically lack the fantastical otherworldly mystique than airships have in fiction. Armored Car (Plane?) Heist
One option might be to have the attackers pull off the airborne equivalent of an armored car heist. This could involve a "pirate" in an armed plane forcing a plane carrying valuable cargo to land, then stealing the cargo at gunpoint on the ground. The attacker, of course, would need to make sure that the plane he targets is still functional enough to land even after being hit. Alternatively, instead of landing himself, he could have a gang waiting on the ground at a pre-arranged location waiting to loot the plane after it's been forced to land. This has actually happened in real life at least once, when a jet fighter commandeered by Taliban rebels forced down a civilian cargo plane shipping weapons to resistance groups in Afghanistan. Aerial Boarding
Another possibility might be to have the attackers intercept the plane and its valuable contents in midair. A look at how this could be done was given in the opening scene of The Dark Knight Rises, where a group of Bane's henchmen jump out of a cargo plane, attached to cables, and break into a CIA plane flying below to rescue Bane. They break Bane out, and then are reeled back into the cargo plane as the CIA plane explodes behind them. The scene in question was actually filmed in real life, without as little CGI as possible, so it is possible something like this could actually be done. Indeed, the idea of snatching something out of the air and reeling it onto a cargo plane goes back decades—the US Air Force used this technique to recover satellite film capsules and drones. Helicopters!
Like airships, helicopters have traits that lend themselves to this trope. They can hover, and while they can't be boarded from another aircraft they can easily be used to deliver attackers to a piracy target at sea. It is easy to picture a gang of pirates using a helicopter to deliver insurgents onto a target ship instead of a small boat. Out of all the methods listed here, this one probably fits most in stories set in the "real" world, in the modern day.