All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The Prophet, sallallaahu ‘alayhi wa sallam, has warned against wearing clothes that hang below the ankle in the Hadeeth narrated by Abu Tharr saying: "There are three (categories of people) whom Allaah will not look at or purify on the Day of Judgment and for them there will be a painful punishment." He, sallallaahu ‘alayhi wa sallam, mentioned among them the one who let his garment reach below his ankles. [Muslim]
Abu Hurayrah narrated that the Prophet, sallallaahu ‘alayhi wa sallam, also said: "Whatever (of the garment) is below the ankles is in the Fire." [Al-Bukhari]
The Ahaadeeth that deal with this issue are so many and can be found in relevant topics included in the books of Fiqh, Hadeeth, and Al-Adab (manners and etiquette). The child (who has not yet reached the age of puberty) is not held sinful for lowering his clothes below the ankle, based on the statement of the Prophet, sallallaahu ‘alayhi wa sallam, which was narrated by ‘Ali : "The pen has been lifted from three," and he mentioned young children until they reach puberty. [Ahmad and Abu Daawood]
The one who should be blamed for that is the one who brings the child a garment that hangs below the ankle or allows him to wear it. Guardians of young children should train them to adhere to the acts of Sunnah and avoid prohibitions, so that they may not get accustomed to them and thus keep committing them when they grow up.
Actually, we would like to thank our dear questioner for paying attention to that matter while many people take it lightly.
Allaah Knows best.