Most authorities rule that contemporary laundry rooms require a mezuzah.[1] 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.[2] They also observe that most people consider automatic washing machines clean enough to be placed in their kitchens and eating areas.[3]
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.[4]
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.[5]
[1] Teshuvos V’Hanhagos 2:648; Leket Shut Mezuzah 4:5, citing R. Nissim Karelitz; Ohalei Yeshurun, citing R. Moshe Feinstein.
[2] Chovas HaDar 2:43.
[3] Birur Halachah, p. 176.
[4] 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.
[5] Birkei Yosef 286:4; Kuntres HaMezuzah 286:107,110; Agur B’ohalecha 32:59; Sha’arei HaMezuzah 18:8.