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Saphire is unlocked near the end of Sakura's "Secrets of Silence" chapter.
I recommend completing Sakura's "Save Sakura" chapter before doing any Saphire quests.
Counsel Saphire during the day
Talk to Saphire when she's in the yard in the day
Go to the Market during the day and help Saphire get a job.
Go to the market
You have unlocked a new way to make money!
Head to the Money-Making page and read Collect Saphire's Rent to learn more.
Check if Saphire is doing well at the market
Talk to Saphire at the market
Ask Agent 17 to hack the market camera
When Agent 17 calls in the evening, give her the task for Saphire.
Punishment & Liar Instructions: Before doing the next quest, if Saphire's Favorability bar isn't filled to 1, go to the Cafe and buy an ice cream. Quickly go give it to Saphire before it melts.
Forgiveness Instructions: Continue to Sakura's "Save Sakura" chapter and come back when you select Queen as your next fate choice or you'll have to do the events of the next quest twice.
Queen Instructions: Give Saphire ice cream from the cafe every day to fill her Favorability bar to 3. Then do the next quest.
Have fun with Saphire in the market
Go to the Market when Saphire is there and talk to her and tell her to go on break.
Forgiveness:
When you select Queen as Saphire's next fate, it'll reset your maximum Favorability to 3. So, you'll have to redo the market scene to get it back to 4.
Queen:
You now have 4 Favorability for Saphire.
Kiss Saphire in the morning
You must continue Sakura's chapter in "Save Sakura" in order to complete this quest.
Have fun with Saphire at night
In the evening when Saphire and Dana are in the Living Room, talk to Saphire and tell her you need to talk to Dana alone. Then go visit Saphire in her room and have fun.