Unlike Ifrit and Titan, who summoned a minion that had to be swiftly destroyed to reduce the damage of their most powerful attacks, reducing the damage from Aerial Blast requires you to keep the stone pillars in the middle of the arena intact. This will require the entire party to work together from the very start of the fight.
The stone pillars take damage from any AoE attacks Garuda does. To mitigate this, the tank should tank her at the edge of the arena (such as where she starts) and face her AWAY from the middle. This will keep Wicked Wheel, Downburst, and Slipstream from damaging the pillars (And the rest of the party). The healer and any ranged DD should attack from Garuda's side, and should avoid standing next to any pillars due to Garuda's Friction attack, which targets a random party member and hits the area around them.
Eventually, Garuda will fly up into the air and come down for her most devastating attack, Mistral Song. She will always land either where the fight began, or on the direct opposite side of the arena. The goal is to place one of the pillars between you and her, in which case you will avoid all damage. If you fail to do this, Mistral Song will most certainly kill you, as it hits for several thousand damage.
Shortly after her first Mistral Song, Garuda will summon several Razor Plumes. These will fly over to the pillars and blow up after a short while (Using a move called Featherlance). Taking these out is the biggest priority for any damage dealer. For most it's fairly straightforward, attacking them with AoE if possible. Arcanist and Summoner have the most difficulty at this phase, as they lack burst damage. However, if they've done a good job of keeping Bio, Bio II, and Miasma on Garuda, they can use Bane to inflict it on most of the plumes, and all three debuffs will likely be sufficient to destroy the plumes.
After two rounds of plumes, Garuda will warp to the center and summon one last batch of plumes to destroy the pillars. These will spawn around the edge of the arena, and so will be more difficult for arcanists to destroy. Once the plumes are either defeated or blow up, Garuda will warp to the center again and use Aerial Blast. As with Mistral Song, players should hide behind pillars to reduce damage. However, the damage reduction is based on the TOTAL number of stones remaining on each pillar, not just the one you're standing behind. This means you can't just focus on protecting a single pillar to survive the blast, and also that there's no difference between hiding behind a 1-height pillar and a 4-height pillar.
After Aerial Blast, Garuda will begin her second phase. She will create a damaging tornado over the entire field, except for a small area around her, essentially decreasing the size of the stage. At this point it's imperative that the tank drag Garuda to one side of the arena so the healer and ranged DD do not get killed by Wicked Wheel.
Her attacks stay the same for the most part with two major changes: Her plumes will now chase after party members and blow up, and can actually be safely ignored (especially if the DDs can self-heal). She will still use Mistral Song, but dodging it is a little different without the pillars. If she warps to the middle, players have to get behind her. If she warps outside the arena, players need to get on the far edge of the calm away from her. It's helpful to note that if she warps outside the calm, she will only ever warp to one spot, where there's a stone "throne" of sorts. So if she doesn't warp back to the middle, just run away from the throne.