I haven't noticed that my squadmates behave differently with different loadouts. On my team, Garrus has a sniper rifle and I still see him engaging at melee range from time to time.
However, I've found I can drastically increase their survivability by ordering them to specific locations. It also helps if those locations are far enough apart that a single grenade won't hit both of them.
(On the Xbox, you do this by pressing Left or Right on the D-pad - one for each squadmate. You can tell them apart by which side their pictures are on in the HUD at the bottom of the display. Pressing "Down" I believe makes them regroup, while "Up" is for them to charge the enemy.)
I tend to balance my team with one "squishy" tech/biotic focused member and one "tank" member with high armor or shields. The squishy squadmate stays close to me, and the tank covers another approach.
For instance, I'll send Garrus off to my right, to a spot where he's got cover on his right and in front of him. Liara I'll keep a little bit closer on my left, and I try to find a similar "L" shaped piece of cover for her. The "L" shaped cover locations are key - this means that if an enemy flanks, they can still stay protected. If you can find some "tall" cover towards the front, that's a bonus - we'll see why later.
I then turn my focus towards the front, where the bulk of the enemy forces are.
If your squadmates go down, I've discovered that if I can get to them, I can pick them back up without using medi-gel. Thus, by staying closer to my squishy pal, I can benefit from her abilities, but I can also save her if she gets hit one too many times. The only downside is that you'll be standing up for this, so if your cover is "chest high" and your shields are down you're going to be soaking up enemy fire. This is where having "tall" cover towards the enemy pays for itself.
If an enemy lobs a grenade at me, I'll run for another bit of cover. This leaves Baron von Squish by herself, but as soon as the grenade goes off, I can get back to the same cover location and pick her back up again.
It's important to pick a location where you're unlikely to be flanked - this can be tough in situations where enemies are jetpacking or climbing rocks to get to you, but through a bit of trial and error in the tougher combat sequences, I can generally pick a spot that's got a few good pieces of cover and not a lot of flanking routes.
The other "gotcha" I'll note is that your squadmates don't always pick the best route going from one cover location to another. If you can, try to stay in one place and avoid advancing until it's absolutely necessary. I've had situations where I wanted to flank up the right side, so I ordered my squad to advance, and they opted to advance straight through the middle rather than circle around the outside, even though their route was longer this way.