Most authorities rule that contemporary laundry rooms require a mezuzah. Firstly, most people no longer launder cloth diapers. Moreover, even if the children soil their clothing, the dirty laundry is only present in the room for a short time. They also observe that most people consider automatic washing machines clean enough to be placed in their kitchens and eating areas.
Your neighbor’s rabbi must be ruling like the dissenting authorities who exempt the doorway of a laundry room from a mezuzah when it is used to store and wash the clothing and diapers of small children.
If there is a chance that the mezuzah will be exposed to children’s soiled clothing, it is advisable to cover the mezuzah with an opaque cover.
 Teshuvos V’Hanhagos 2:648; Leket Shut Mezuzah 4:5, citing R. Nissim Karelitz; Ohalei Yeshurun, citing R. Moshe Feinstein.
 Chovas HaDar 2:43.
 Birur Halachah, p. 176.
 Shevet HaLevi 2:156. Agur B’ohalecha (32:47:109) asserts that in a later teshuvah (5:166), Shevet HaLevi changed his ruling in later years when cloth diapers were no longer laundered in the home. Kuntres HaMezuzah (100) rules all laundry rooms are deemed undignified.
 Birkei Yosef 286:4; Kuntres HaMezuzah 286:107,110; Agur B’ohalecha 32:59; Sha’arei HaMezuzah 18:8.