Is there any upgrade that might help stop being poisoned, or cause it to do less damage?
There are not any foundry upgrades, perks, or gene mods that provide poison immunity, but you can research several armors and items that provide it
In the base game, the Titan Armor provides poison and fire immunity. In the Enemy Within expansion, The Titan Armor, Ghost Armor and Archangel Armor all provide poison, fire, and strangulation immunity.
MEC soldiers are also immune to fire and poison, but MEC troopers are only available in the Enemy Within expansion.
So I should just outfit my whole squad with poison immunity and steamroll everyone, right?
Poison immunity is nice, however enemies will change their tactics if they are unable to poison anyone. This can make it worse for your squad than just allowing someone to get poisoned, especially if your squad is not in the late game armors with extra health. As specifically discussed on the Respirator Implant wiki page:
If all squad members are immune to poison, Thin Men are certain to use their Light Plasma Rifle instead of Poison Spit. This risks a critical hit for up to 10 damage, compared to Poison Spit's guaranteed 3.
And since poison immunity usually comes with strangulation immunity (in Enemy Within):
If all squad members are immune to strangulation, Seekers will use their plasma weapon instead. This can deal up to 6 damage, compared to the strangulation's guaranteed 2 damage.
So feel free to use poison immunity, just be aware of the trade-offs it provides.