The case for baby ear piercing is real: in many families and cultures it's a cherished tradition done in the first months of life, babies don't anxiously anticipate the pinch the way older kids do, they can't reach up and twist new earrings the way toddlers famously do, and a parent controls the aftercare completely. Pediatric guidance in the US generally treats piercing as safe at any age when done cleanly — while often suggesting parents consider waiting at least until initial infant vaccinations are underway, and some pediatricians prefer waiting until a child can care for the piercing herself.
The case against comes largely from within the piercing profession. Many members of the Association of Professional Piercers and similar high-standard piercers decline to pierce pre-verbal children as a matter of ethics: an infant can't consent to a permanent body modification, can't say where it hurts if something goes wrong, and her earlobe will keep growing — a stud placed dead-center at four months can sit noticeably off-center at fourteen years.
Neither side is being unreasonable, which is why this decision belongs to informed parents. What both sides agree on completely: if it happens, it should happen with a needle in expert hands with implant-grade jewelry — not with a gun at a retail counter.