Crafting jobs each have a crystal type every recipe uses, then a secondary crystal that only some of the recipes use:
Carpenter - Wind, Ice
Blacksmith - Fire, Earth
Armorer - Ice, Earth
Goldsmith - Wind, Fire
Leatherworker - Earth, Wind
Weaver - Lightning, Wind
Alchemist - Water, Lightning
Culinarian - Fire, Water
Looking at the recipes, seems primary crystals are all that's needed for crafting recipe components, but the final product recipes typically require both crystals. ie One crystal to turn iron ore in to an iron bar, both crystals to turn that iron bar into a sword.
All crafters have an ability called Byregot's Blessing (Astral Lightning, god of crafters), which provides a boosted increase in quality at the cost of Inner Quiet stacks.*
Gatherer classes each currently have a god-related ability (whose elements kinda correspond with their guilds' home cities, though technically Limsa has a wind patron but it's also on the water so...):
Botanist - Nophica's Tidings - Astral Earth - increases item yield
Miner - Nald'Thal's Tidings - Umbral Fire - increases item yield
Fisher - Thaliak's Favor - Astral Water - Restores GP at the cost of Angler's Art (fishing crit) stacks.
Also, MIN and BTN have a passive trait called "Nymeia's Ward" (Umbral Water, fate goddess) which increases item and crystal yield at nodes.*
As for Astral/Umbral:
With crafters, you could see the two element combos as the primary crystal being the required actor in the synthesis, and the secondary element tempering the craft when necessary. So, kind of astral (active) and then umbral (passive)? But I don't think any NPCs or codices state this outright. Its just an observation of mine.
With gathering, the gods' have umbral/astral aspects to them if you look at the months they represent, but I'm not sure if that really conceptually effects the boons. I think the gathering ones are more about the things the gods are patrons of or their overall elements. I listed them anyway, though.
*These are just the current builds, of course. In older builds, I think crafters and gatherer's may have had more Twelve related abilities. Dunno where we'd find that info archived, though.