Dog and newborn

What do people think about dog fur with a newborn? We have a two year old lab who loves our 3 week old baby. Our dog never licks her and just sniffs her. When she cries, he goes over and looks at her to make sure she’s okay. My mom (who is graciously helping us) keeps freaking out about the dog fur that happens to get on the baby’s clothes and other things. I know it’s important to have things be clean but she overly stressing me out with her almost OCD like tendencies.