I have had several legitimate arguments with my friends about this.
My conclusion: It's all about the timimg. Depending on the time in which both of the universes meet, it could result in either sides' victory because of political instability or combative exhaustion. However, for the sake of argument, let's say that both have an amalgem of all the different factions and individuals taking up arms to side against the opposite force.
Begin wall of text.
Let's start with the similar forces. For almost everything in Star Wars, there is a tactical match in Mass Effect and vice versa. The opposite of the Republic would be the Alliance. Comparible for obvious reasons, but the Republic has far more resources and troops than the humans. However, there are also the Batarians, Turians, Quarians, and Elcor. Combined, the forces of the Clone Army and the combined forces of the various "troop" races in Mass Effect, the fight would be very even. This is because neither side would have good defenses against the others' weaponry. While Star Wars has a strong plastoid composite for armor to defend against plasma bolts, Mass Effect soldiers don't use plasma weapons, except for a few exceptions. (I'll talk about this later.) Similarly, Mass Effect soldiers have few defenses against plasma weapons, instead having ablative ceramic hardsuits designed to survive impacts against high-velocity projectiles. The deciding factor is the shields; Mass Effect troopers have them, Clone Troopers don't. This small advantage gives the Alliance a somewhat viable, if not hugely effective defense against beam weapons.
Now, some more specific troop comparisons. Wookies are some of the most iconic Star Wars beasts. They find a match in the brutish krogan of Mass Effect. Both are ludicrously strong, very tough, and are veritable tanks compared to infantry. Also, both species are very familiar with melee combat, wookies ripping the arms off of their victims, the krogan employing a feral rush to stun and incapacitate the enemy. In the end, it comes down to range. If they meet on a larger battlefield, the wookies would win due to their longer range bowcasters. However, if they met on even a remotely close quarters battlefield, the wookies would be demolished by the krogans' high survivability and shotguns.
Jedi. If no one has wondered what their comparison is, here you have it. The matter is very simple; biotics. To make a long story short, biotics would win. Why you ask? Jedi are so powerful aren't they? What could biotics possibly do? Here's the answer. The Asari. Every single one is biotic. I'm not saying the entire species would wage war, but the number of combat-ready asari massively dwarfs the number of jedi. There are something like 300 jedi, and well over 1,000,000 full-fledged, combat-hardened biotics. (Not just asari, though.) It would be a war of attrition. For every jedi (or sith) that falls, they would have to kill 30,000 biotics. Talk about overwhelming odds.
Now, my favorite part. Droids would match up with geth. It would definitely be a close call in this fight, and I'd have to call the fight a tie. Although the droids have tons and tons and tons of troops, the geth are simply more adaptive. Destroyers are better than Super Battle Droids, Snipers, Rocket Troopers, and regular Troopers are better than the droid infantry, especially in massive groups. The armatures would fight the Spider droids, the Primes would fight hailfire droids. Overall, a very close and epic battle.
Now, for some vehicle action. Star Wars would definitely win in this case. It has a whole plethora of different ground vehicles, most notably the AT-AT and AT-TE. While the Makos and Hammerheads could do some damage, they would be absolutely annihilated by the firepower of the opposition.
Finally, and most importantly, space combat. In the Mass Effect would win. Why? Just a collection of seven letters would vindicate Mass Effect's victory; Reapers. If the end cutscene is anything to go by, there are a LOT of them. Thousands, maybe millions. They could take down the fleet of any faction in Star Wars with very little trouble at all. Plus, they ahve the Alliance fighters, cruisers, dreadnoughts, frigates and heavy cruisers for supporting fire. In the end, no contest in the space category. And for all those people thinking "the Death Star would do a ton of damage!", you are wrong. The Death Star needed hours to calibrate its main gun on a PLANET, let alone a fleet of ships.
Also, last but not least, the creepy crawlies of the two worlds. It would be the Yuuzhan Vong versus the collectors, and other creepy beasts(i.e. rancor, acklay) versus scions, praetorians, and husks. It would be close but once again, Mass Effect would win. Harbinger, Guardians, Assassins, and Drones would out manuever the Yuuzhan Vong, and destroy them. I don't want to go too much into tactics here, because I don't know very much about Yuuzhan Vong, but I think they would lose. The beasts of Star Wars would thrash the Reaper experiments, as no force could take on multiple rancors and acklay at once, end of story.
Sorry again for the absokute wall of text, but in the end Mass Effect would win mainly because of their more diverse trops, tactics, and weapons.
/thread.