Upon reaching level 64, you may want to visit a Consortium Recruiter in Nagrand and get [66] The Consortium Needs You!. This is only worth 10 reputation, but is not available if you've completed any other Consortium quests (such as the repeatable ones below).
While visiting Aeris Landing in Nagrand, accept both [66] A Head Full of Ivory (3 [Pair of Ivory Tusks]) and [66] Stealing from Thieves (10 [Oshu'gun Crystal Fragments]). Pick one of these to complete, then complete the repeatable version of the one you picked until you reach friendly (12 total turnins including the first). At that point, turn in the other quest -- doing this will provide one additional turnin that would otherwise be unavailable later.
Dungeon runs of Mana Tombs provide reputation also. This can be a good option if you don't want to spend time or money completing the repeatable quests above.
Characters trying to simultaneously earn reputation with the Mag'har or Kurenai and the Consortium may want to focus on killing Ogres in Nagrand, which award 10 Mag'har or Kurenai reputation per kill, and saving the [Obsidian Warbeads] for Consortium turn-ins. Doing this also nets many [Oshu'gun Crystal Powder], for [Halaa Research Token] turnins, allowing a player to effectively farm their Nagrand faction rep, Consortium rep, and half of their Halaa tokens all at the same time. The only caveat is the drop rate, which is roughly 33% for the warbeads, while it is 50% on the insignias.
If you are level 70, want a faster grind without concern for Mag'har/Kurenai reputation, and can handle the slightly more powerful Ethereals, then you may want to grind insignias instead.
Open Ethereum Prisons with [Ethereum Prison Key] and kill the mob that spawns to turn in [Ethereum Prisoner I.D. Tag]. The mobs most likely to drop Prison Keys typically drop very few items other than gold, netherweave and green items-- which allows you to grind for a very long time before your bags fill up. In patch 2.3, [Ethereum Prison Key] is now easier to obtain as it is a reward for the daily non-heroic dungeon quest.
Complete the "daily dungeon" quests from Wind Trader Zhareem and Nether-Stalker Mah'duun. These appear to provide consortium rep, despite the NPCs not being marked as Consortium members (350 rep for Heroics, 250 for Non-Heroic).
Upon reaching Friendly status, players are officially considered members of the Consortium and given a salary. The salary is a bag of gems at the beginning of every month, given by Gezhe at Aeris Landing. Higher reputation with the Consortium yields better and more jewels each month.
Note: Due to a bug, no gem payments were received between the expansion's release and the patch 2.1.0, where it was fixed. However, players will receive their gems by talking to Gezhe instead of through the mail. According to Blizzard poster Pavonum, there is no plan to reimburse players for missing gem payments. [1].