Asare studied Fine Arts at the College of Art in Kumasi, and between 1967 and 1979 he was a teacher in Ghana. During this time he began to write and illustrate children's books, including the much translated Tawia Goes to Sea, which also received the UNESCO citation, "Best picture book from Africa."
After a period of ten years wherein which he could not publish any work, Asare returned in 1981 with a new book, The Brassman's Secret, which was translated into many languages, and won the Noma Award in 1982 as the best book published in Africa in the preceding year.
Many further successes followed. Asare moved to London in 1983, and this has been his base ever since, though he frequently travels through Africa, looking to experience as many African cultures as possible so that he can represent them in his works. His book, Sosu's Call, was the winner of the 1999 UNESCO First Prize for Children's and Young People's Literature in the Service of Tolerance.
- Noma's Sand: A Tale from Lesotho (Children's Literature, Sub Saharan Publishers, 2002)
- Meliga's Day (Children's Literature, Sub Saharan Publishers, 2000)
- Nana's Son (Children's Literature, Sub Saharan Publishers, 2000)
- Sosu's Call (Children's Literature, Sub-Saharan Publishers, 1997)
- The Magic Goat (Children's Literature, Sub Saharan Publishers, 1997)
- Halima (Children's Literature, Macmillan, 1992)
- Bury my bones but keep my words: African tales for retelling (Short stories, Harper Collins, 1991)
- Cat in search of a friend (Children's Literature, Jungbrunnen, 1984)
- Chipo and the bird on the hill: A tale of ancient Zimbabwe (Children's Literature, Zimbabwe Publishing house, 1984)
- The Brassman's secret (Children's Literature, Education Press, 1981)
- Tawia goes to Sea (Children's Literature, Ghana Publishers, 1970)
- Mansa helps at home (Children's Literature, Ghana Publishers, 1969)
- I am Kofi (Children's Literature, Ghana Publishers, 1968)