There's a selection in the kiosk specifically for verifying your passport in your profile. Simply checking in with your passport is not sufficient (it's just using your passport to get your name and look up your itin). I don't recall exactly where the selection is in the kiosk, but I do recall it is separate from the check-in process. I did it for both myself and my wife last year and our account profiles now show "Verified By Delta" with our passport info.
A separate popup will ask do you wish your passport to be saved. Select yes, and you should be good to go.
I find that it sticks as long as I don't checkin with an agent. Once I do that on a future trip most times the passport gets deleted.