Reputation with Ve'nari in Shadowlands is increased by killing rare enemies and completing quests.
Ve'nari has more than his fair share of items to sell, but they're all stuck behind the equalization of his reputation. There are a handful of ways to increase your reputation with her in Shadowlands, but all of them will increase your Eye of the Jailer counter. At rank 5, you will no longer gain reputation or stygia (the currency used to purchase Ve'nari's goods), so you need to have a plan of action every time you get knocked down.
The most effective way to increase your reputation with Ve'nari in Shadowlands is by completing quests. At first, she will have a few story quests to complete, but eventually you'll need to increase your reputation with her to unlock more. To get started, take them out, then focus on the daily quests that appear in the Maw. Eventually, you will increase your reputation with Ve'nari up to the next rank, in which case additional story quests and weekly quests will be available.
Weekly quests reward the most reputation, but they also boost your Eye of the Jailer up a rank after you complete them. These are followed by daily quests, then random "lore" quests that can drop both regular and rare enemies in the Maw. Speaking of rare enemies, they also reward Ve'nari's reputation. They also increase your Eye of the Jailer counter. The rarer the monster, the more reputation you will gain, with the rare common “special encounters” offering the smallest amount, and the harder to spawn the more.
Below is a table from Burningbamf on WoWHead that displays all sources of reputation, the amount of reputation gained, the amount your Eye of the Jailer meter will increase (each rank takes 1000 points to achieve), and the overall amount of Reputation earned by eye point. Bottom line: Focus on your quests first, then start hunting for rare spawns and other sources of reputation. Once you've hit Eye Rank 5 you're done for the day, so get in the Maw with a Blueprint and engage.
