Blast Assault works pretty well, as it does a ton of damage per shot which means you don't have to remodulate as often. Miniguns and dual pistols are among the worst, as they fire a ton of shots and only do a little damage per shot.
Pulsewaves are the most powerful anti-borg weapon in the game. If you are a science officer, then carry tachyon harmonics (currently broken with season seven), photon/plasma grenades or lunge as tactical, and mines/mortars as engineering. Science officers simply strip away borg shields and tactical/engineering officers deal kinetic damage to get around the powerful borg shields. As said above, swords are powerful, but it takes time killing borg with swords. Pulsewaves are the bread and butter of Special Task Force missions. Their high damage at close range takes down borg very fast. A good secondary is a sniper/split beam/high density beam rifle.
In STFs, melee weapons are a good way to get yourself killed or assimilated.
Of the guns available, I'd say Pulsewave and Sniper are your best shots. Intelligent use of their knockback capabilities can make certain missions a lot easier (knock the drones into the plasma in Rebecca's room in Infected, for example), and Sniper rifles allow you to deal decent damage even when far away (again, such as when fighting Rebecca).
Weapons I have found that work well on the Borg (provided you don't have STF gear):
Weapon 1: Tetryon Sniper Rifle, Polaron Sniper Rifle, Phaser Sniper Rifle
Weapon 2: Phased Tetryon Pulsewave Assault (you just can't beat it when it comes to drone slapping)
Weapon 1: (see above)
Weapon 2: Shard Sword of Kah'less, Phased Tetryon Pulsewave Assault (Shard sword is epic enough that it can be used to replace the pulsewave)
However in all honesty, the KHG Pulsewave and MACO Battle Rifle are probably the best weapons for fighting the Borg. I am not sold on the Omega Carbine... But I have been told it's nasty as hell. I have also seen the Borg adapt very quickly to it.
Just my 2 ECs...
It is said the best weapon is one that is never fired. I disagree. The best weapon is one you only have to fire... once.