James has Carnage, which can kill a lot of enemies in one hit but recharges very slowly, especially if he has activated Fortification. You can try repurposing him as a sniper by giving him the M-99 Saber, though this makes it all the more important that you switch him to his shotgun when necessary. James doesn't have a great shield-stripping ability, but he brings excellent durability and good weapon damage. He is the best bait for enemies like Banshee or Brutes.
Ashley. I recently started a game with her in it for the 1st time. She has decent crowd control with Concussive Shot and Inferno Grenades, as well as a sniper rifle. She also has rapid fire capabilities from her assault rifle, but she's less durable than James and Kaidan. You'll want to emphasize sniper rifle bonuses in her build, so that she will tend to stay in cover. Disruptor Ammo upgraded for stun and for sharing with squad members will give your whole group some anti-shield and light crowd control.
Kaidan's powers let him counter shields, barriers and armor and he provides decent firepower if he uses his assault rifle. With help of Barrier and getting his cooldown reduction costume to counter Barrier's penalty not only he deals highest damage among biotic squad members, he provides substantial bonus power damage to other squad mates as well, making him the best to take with squads focused on power usage. Thanks to Reave's protection buff and Barrier he is the 2nd tankiest squad mate after James.
Liara lacks firepower and durability, and some of her powers are devalued by shields. But Singularity and Stasis all but guarantee a dead enemy, especially to a player with good aim. Don't Evolve warp's damage! Warp with the Expose evolution will help against the many armored opponents. Re-spec her Stasis with bubble to be able to grab even shielded enemies from behind cover. A biotic Shepard can combo with her Warp and other powers to create instant and frequent biotic explosions even against shielded and armored enemies. Liara's damage is low but due to low cooldowns and good crowd controls she is an OK squadmate especially with adept shepard.
Garrus has similar weaknesses to Ashley, in that he either hits slowly or exposes himself to oncoming fire, but with a fully upgraded sniper rifle he can do substantial damage, even on Insanity. Micromanaging his equipped weapon choice is critical--either that, or have a third member who can respond quickly while your sniper is reloading. Overload with the Chain and Neural Shock upgrades provides crowd-control and anti-shield capabilities.
EDI, like Kaidan, can handle both shields and armor, but lacks some of his tankiness and firepower. Her Decoy can be deployed for crowd control, particularly with groups of small enemies. She can prime both fire explosions and tech bursts if the team has a generic detonator such as Concussive Shot or Carnage. If you bring her, make sure to max her Defense Matrix and keep it active, thus providing her with decent durability.
Tali has decent crowd control and damage output via drones and shotguns, but their short range exposes her durability issues. If you bring her, make sure to max out her Energy Drain.
Javik comes with decent firepower from his assault rifle. This is especially true if you've equipped him with the Particle Rifle. His Lift Grenades will provide substantial disruption of groups of lesser enemies, and he can prime multiple biotic explosions with Dark Channel. Because a single cast of Dark Channel will jump to a new enemy if the first one dies, a Shepard with the fast-recharging Throw will find Javik to be a valuable squadmate. Javik is unable to burst a target down which is his main weakness, he deals damage gradually.