Putting babies in cradles is permissible in the light of the Shariah. There is proof of this in the Holy Quran and the Hadith. The Holy Quran has mention of Hadhrat Isa (Peace be upon him) in a cradle. It is given in Surah Maryam:
They said: "How can we talk to one who is a child in the cradle?"
Surah Maryam (19:29)
He shall speak to the people in the cradle
Surah Al-E-Imran (3:46)
Also, it is well established that the Holy Prophet (Sallallahu alaihi wa sallam) also has rested in a cradle. Imam Suyuti (May Allah shower His Mercy on him) has recorded a Hadith in Khasaais Kubra on the authority of Baihaqui, Sabuni, Khateeb and Ibn Asakar