South almost never checks anything whatsoever so it's a moot point there unless you're crossing in a major city like Tijuana. On many occasions even the guy at the metal detector has been sleeping. (Also note: Mexico never changed law like US, in Mexico it's still legal to enter with only a birth certificate and ID)

Going North into america I'd say the majority don't have passports and they are let through anyway. I actually have a legal US passport and I watched myself detained, jailed, and interrogated while everyone without a passport was let through with a birth certificate and state driver's license (this is modern times, not the days when that was all you needed). So maybe it is worse to have a passport as it's suspicious as most people at the lesser land crossings do not.