This came up a few months ago, and I wasted most of a day figuring it out. Here's a compilation of what I said at the time:
The Borg's main advantage is its ability to counter energy weapons. They do this with the Adaptive Shield Matrix:
All phasers are generated on a particular subspace phase compression pulse frequency, whilst torpedo warheads all possess their own shielding which also possesses its own subspace phase compression pulse frequency. Adaptive Shielding works by remodulating the shields to the identical subspace compression pulse frequency of torpedos and phasers...
So, it seems like the Borg's sheilds adapt to the subspace pulse frequencies of phasers and torpedos. Phasers are a type particle weapon that Gene Roddenberry made up when he realized lasers didn't work the way he thought they did, and they don't really have much basis in the real world like lasers or plasma weapons do. The problem is that 99% of the weapons in Star Wars are plasma weapons, except for the Death Star cannon, which is a laser.
Since the Adaptive Shield Matrix specifically works by adapting to subspace frequencies, there's really no reason to think that their shields would have a distinct advantage over lasers or plasma bolts the way they do phasers. Star Fleet seems to agree, as they theorized plasma phasers would be effective weapons against the Borg in Best of Both Worlds. Picard was also able to easily kill several Borg with hard-light bullets in First Contact, further demonstrating their inability to counter non-phaser weapons.
It also doesn't seem like the Borg have an innate ability to adapt to enemy weapons. Aside from only countering phaser weapons, the rotating-frequency strategy was pretty effective throughout TNG and First Contact. It seems more like that Adaptive Shield Matrix was just a piece of technology they assimilated rather than an intrinsic ability to counter attacks.
So, if the Borg shields don't nullify the Death Star's weapons like they do Star Trek weapons, and they don't have a special ability to counter enemy attacks, this just comes down firepower. The superlaser should be able to destroy any Borg cube multiple times, and even without the superlaser, they're massively outgunned. This is 3000 meter ship against a 75 mile wide battle station. Even if the 10,000 turbolaser, 2,500 laser cannons, 2,500 ion cannons can't overpower them (and by the way, it sounds like those, "lasers," are actually plasma weapons according to wookiepedia, because of course Star Wars can't be consistent), and the 768 tractor beam projectors can't immobilize the cube, the 7,000 individual tie fighters would probably overwhelm the it. Hell, if the Death Star is faster, they could probably just smash into them and still survive the damage.
I think the Death Star has this by a mile. I hate to admit it, but I don't see a win condition for the Borg here.
The Borg would want that technology so after that first cube is destroyed they'd be back with enough firepower to overwhelm the shields and get drones on board. Even if they lost a couple of hundred cubes it would likely be worthwhile.
The Borg have several trillions of drones many thousands to millions of vessels. They can go full Russia tactics with capturing the death star if needed.
If this is the first death star though, they'd just beam some drones aboard while it was busy destroying them.
There were small gaps in the shields of the first Death Star; the Empire believed that this was harmless since only small ships could wiggle through the gaps.
If a small ship can get through then a transporter beam is definitely getting through.
Yeah, the Borg hordes would overwhelm the Death Star. The last time this was posed was as a single Borg Cube vs. the Death Star, so I'm assuming that's what OP was asking.
I don't really see how a boarding party is going to help them much, though. Stormtroopers aren't the most competent soldiers, but Borg aren't going to be better protected from the plasma in their Blasters than the Cube is from the Death Star's plasma and laser weapons. It's possible they could assimilate someone with strategic knowledge of the station or find something useful in their computers, but they're going to get wrecked before they can pull that off. The cube is also too big to get through the Death Star shields, so they can't use the exhaust port trick to blow it up.
I don't see how plasma weapons would do anything. The Borg are probably already adapted to be honest. The Romulans were rocking plasma torpedoes in the 23rd century, and there's no idea of just sending a bunch of TOS era warbirds against the Borg. The only reason the primary weapon would work is the raw power it has being able to simply overload anything in front of it.
If the Borg personal shields aren't already adapted to storm troopers weapons they'll just blast through a few Borg, then they adapt their personal shields to compensate for the stormtrooper weapons.
After that it's open season on turning the whole crew into drones and subverting the computer systems.
The only thing that might be able to stop them if there was a force wielder on the death star, and they realised what was going on before even they'd be overwhelmed by the sheer number of assimilated crew.
Well, again, their sheild adapting technology very specifically works on phaser and torpedo warheads, but haven't shown any adaptability to other weapons, and the entry on Memory Alpha entry seems to make this explicit. That doesn't mean that they don't stop other weapons, just that they only work like normal shields against other weapons. So, 23rd century warbirds probably wouldn't be very effective against the Borg, since they're old, outdated, and probably underpowered tech, but a Galaxy Class Starship with plasma weapons as powerful as normal phasers would probably do well.
So, if the sheilds don't nullify the Death Star's weapons, this match up would basically be the same as any large Star Trek ship vs. the Death Star. And since the Death Star is so massive, pretty much any ship is gonna get wrecked. I'm sure the Borg would do fine against a Star Wars ship of roughly comparable size, but the Death Star is almost 100 times larger.
Not sure where you got in memory alpha that they can only adapt to phasers, this is what memory alpha has to say.
The Borg were usually exceedingly quick to adapt; their shields would often nullify nearly any energy weapon, and their weapons could usually penetrate nearly any shield or defense, within minutes
Oh, you're right, it's the Borg Wiki, it just looks the same because they both run on Fandom. Still, there's no reason to think that's no longer cannon just because Destiny is no longer cannon. We never see borg weapons adapting to anything other than phasers and photon torpedos, and there's lots of circumstantial evidence that they can't adapt to other weapon types.
There's also circumstantial evidence that they can adapt to other weapon types. Klingons and Romulans both ice disrupters and they're are drones of those species. But probably even more telling, star fleet never whacked plasma cannons on their new anti Borg ships, when it was a well established technology. On personal shields I'll grant you that they seemed to struggle (or maybe it just wasn't used enough) to adapt to holomatter based projectiles.
For the ships themselves I grant you the raw power and tonnage will tell. But for a Borg boarding party I see no reason for storm troopers to fare any better than others.
I mean, after being shot through by the death stars laser weapon, the borg would launch their time travel device so as to go back and include some ridiculous vulnerability in the death stars plans that they could then take advantage of. Say, like a single vent shaft that ran from the outside of the fort all the way to it's power source 🤣 I also imagine they would be able to get drones aboard. however, I could also see the hivemind being vulnerable to force manipulation tactics. A larger consciousness might be harder to sway, but the power of Darth Vader can't be summarily dismissed in this capacity. I think Darth would attempt to control them.
Yeah, I suppose time travel could win them the day, but that feels like cheating. I'm not sure boarding party would do much better. Those personal shields the Borg have are...weird. They stop phaser fire like a ships shields, but they don't stop physical objects (for example, Mr. Worf's fists) like a regular shield. But there's no reason to believe that they would stop plasma bolts from a Blaster, so there's no reason to think a Borg boarding party would get wrecked by a barrage of (admittedly inaccurate) Stormtrooper fire before they could do much damage. Plus the Cube would probably get destroyed before they could send more than one or two waves of drones over.