I think higher luck will increase your chance of non Legend/Eternal items having a Crystal affix. in the time it takes me to go from using one Legend map to getting my next Legend map, I have gotten 0-3 items with a Crystal affix. I don't include Legend/Eternal items here because they seem to have a lower chance to have a Crystal affix as a normal item. my luck is at 550 for now and working good enough. also, I am on mythic 3 difficulty, which also gives me a luck boost, with the Fortunate Perk. well, now that I look at it, that is where all my luck is coming from.
and as @dickwad said, you can use the Obsidian crystal to put 1 permanent affix on an item.
if you use dust to get Legend/Eternal items from the Codex, they have a chance to have a Crystal affix too.
I can't remember if items bought in the shop have a chance to have a Crystal affix.