Many people have a view of probiotics being a magic bullet that can do no harm and are safe for a 3 month old or a 99 year old. This post attempts to document what we know about the use with infants. The short of it is that there are significant risks.
Probiotics use has been grandfathered in without requiring safety tests in the US.
” The use of probiotics cannot be considered risk-free and should be carefully evaluated for some patient groups. “Infectious complications following probiotic ingestion: a potentially underestimated problem? A systematic review of reports and case series. 
infections caused by lactobacilli, mainly endocarditis, bacteremia, and pleuropneumonia, occasionally occur. The relevance of Lactobacillus spp. and other members of the LGC as opportunistic pathogens in humans and related risk factors and predisposing conditions are illustrated in this review article with more emphasis on the species L. rhamnosus that has been more often involved in infection cases.Members of the Lactobacillus Genus Complex (LGC) as Opportunistic Pathogens: A Review. 
- ” Three patients were given a diagnosis of B. longum bacteremia: 2 patients in a neonatal unit in which 17 patients were given oral probiotics and 1 patient in a neonatal unit in which 31 patients were given oral probiotics… Thus, the incidence of Bifidobacterium bacteremia is theoretically underestimated. ”  So risk of developing bactermia is 3/48 or at least 6%
- Are probiotics safe? Bifidobacterium bacteremia in a child with severe heart failure. 
- necrotising enterocolitis (NEC) – NEC was significantly reduced in probiotic group infants fed any breastmilk [20/179 (11.2%) vs. 10/183 (5.5%); P = 0.027]. No benefits were noted in exclusively formula‐fed infants [4/54 (7.4%) vs. 6/44 (13.6%); P = 0.345] (Repa et al., 2015). 
- Probiotic sepsis: The reports of probiotic sepsis and the death of one preterm infant due to fungal sepsis from a contaminated probiotic product justify the concern about probiotic supplementation in preterm infants (Centers for Disease Control and Prevention, 2014; Bertelli et al., 2015; Esaiassen et al., 2016).
- ” This is the largest trial to date of a probiotic intervention. It shows no evidence of benefit and does not support routine use of probiotics for preterm infants. “
- Prophylactic administration of B.clausii to preterm neonates did not result in a significant difference in the incidence of LOS as compared with placebo. 
- Our observational data support the use of Lactobacillus acidophilus/Bifidobacterium infantis probiotics to reduce the risk for gastrointestinal morbidity but not sepsis in very low birth weight infants. 
- Genomic and epidemiological evidence of bacterial transmission from probiotic capsule to blood in ICU patients. 
- Meningoencephalitis caused by Lactobacillus plantarum – case report. 
- Breaking Bad: a case of Lactobacillus bacteremia and liver abscess. 
- Probiotic related Lactobacillus rhamnosus endocarditis in a patient with liver cirrhosis. 
- Mitigating risk of bloodstream infection related to inpatient probiotic use. 
- Are Probiotic Really Safe for Humans? 
- Lactobacillus: the not so friendly bacteria. 
- The evidence for benefit of probiotics in infantile colic is strong but limited to the L. reuteri DSM 17938 strain and to breast-fed infants. 
If there is a risk/indications of necrotising enterocolitis or sepsis for an infant, studies indicate that the risk/benefit for an infant is improved with probiotics [2018, 2017]. ” Introduction of probiotics was associated with a reduced adjusted odds for ‘NEC or sepsis or death’ in exclusively breastmilk-fed infants (OR 0.43, 95% CI 0.21-0.93, p = 0.03) only. ”
My advice is to not use probiotics for a child or in fact unless there is a serious health risk that the right probiotics have been documented to actually help.
Any probiotic claiming that it is designed for children — demand that they share the clinical studies for their formulation. They are likely pissing urine to make sales (excuse my language). If you get such a study and have a sick child — you really should be asking for studies with similarly sick children.
It is fine for an adult to make a decision to take or not to take; but with a risk of sepsis being established with no certain benefit – IMHO, it’s inappropriate for a parent to make that decision for a child.