The story of the Moses basket comes from Exodus 2:1-10. In it, Moses is left in a basket amid the reeds and bulrushes along the Nile, in Egypt where the Hebrews had been living since the time of Jacob.
By the end of Exodus 1, the pharaoh of Egypt (perhaps Ramses II) had decreed that all the Hebrew boy babies were to be drowned at birth. In the 10 verses from Exodus 2, Yocheved, Moses' mother, hides her newborn for 3 months and then places her baby in a caulked wicker basket in the Nile River reeds. The baby's crying alerts one of the pharaoh's daughters who takes the baby. Moses' sister Miriam watches in hiding, but comes out when it is clear the princess is planning to keep the child. She asks the princess if she would like a Hebrew midwife. The princess agrees and so Miriam arranges to have the real mother get paid to nurse her own child who now lives among the Egyptian royalty.
Source: Biblical Literacy, by Rabbi Joseph Telushkin.
Here is the Exodus passage:
Exodus 2 (World English Bible)
1 A man of the house of Levi went and took a daughter of Levi as his wife. 2 The woman conceived, and bore a son. When she saw that he was a fine child, she hid him three months. 3 When she could no longer hide him, she took a papyrus basket for him, and coated it with tar and with pitch. She put the child in it, and laid it in the reeds by the river's bank. 4 His sister stood far off, to see what would be done to him.
5 Pharaoh's daughter came down to bathe at the river. Her maidens walked along by the riverside. She saw the basket among the reeds, and sent her handmaid to get it. 6 She opened it, and saw the child, and behold, the baby cried. She had compassion on him, and said, "This is one of the Hebrews' children." 7 Then his sister said to Pharaoh's daughter, "Should I go and call a nurse for you from the Hebrew women, that she may nurse the child for you?" 8 Pharaoh's daughter said to her, "Go." The maiden went and called the child's mother. 9 Pharaoh's daughter said to her, "Take this child away, and nurse him for me, and I will give you your wages." The woman took the child, and nursed it. 10 The child grew, and she brought him to Pharaoh's daughter, and he became her son. She named him Moses, and said, "Because I drew him out of the water."