If someone crosses the US/Mexico land border and has no i.d. what are their options?
As a minimum likley refusal to enter the US unless secondary inspection indicates you have a relevant visa or are a USC where you will then be paroled. At worst you get put in jail, deported and banned from re-entry especially if you are not truthful to the inspector including a false representation/ declaration of US citizenship.
If they are a LPR and have no i.d. how can they prove their status?
DHS have your details on file including biometrics as relevant. Secondary inspection will reveal the truth. See my comment above.