Morobe was born on 2 October 1956 in Soweto. When he was young he spent a lot of time with his uncle, a Minister in the African Methodist Episcopal Church, and moved around with the family. He began school in Ermelo, but in 1996 returned to his parents in Soweto. He completed Primary School in Soweto and went on to Orlando North Secondary School and Isaacson High School. In high school he became increasingly interested in politics and history, and he began to read up on them at libraries.
In 1972 he joined the South African Student’s Movement (SASM) because he saw unity and community development as important. He was also starting to think about doing something against the education system and apartheid as a whole. The group was very influenced by Black Consciousness ideology, and also organised cultural activities. In 1973 many members of the committee were detained, and it became quite weak. In 1974 he helped with the reorganisation of SASM, and was elected treasurer. In 1976 SASM was to play an important role in the Soweto Uprising, and it was the first time that Morobe took part in a demonstration.In August 1976 the Soweto Students' Representative Council (SSRC) was formed and Morobe became deputy Chairman. The group played a role in organising campaigns and protests, and almost all Soweto schools were involved in the SSRC.
In December 1976 Morobe was arrested for the first time. He was moved around from prison to prison, kept in solitary confinement, asked to testify against the ANC, tried for conspiracy to commit sedition and eventually sent to Robben Island. He was released only in May 1982, and by this time had completed thee matriculation subjects.After his release, Morobe returned to politics and joined the Congress of South African Students (COSAS) and served on the Executive of the Soweto Youth Congress. He also became involved with the General and Allied Worker’s Union, and it was through this that he participated in the formation of the United Democratic Front (UDF) in 1983.
Morobe became very active in the UDF during the 1980s, and served on the Executive. From October to December 1984 Morobe was again detained. During 1984 Morobe left the county to approach the British government and the United Nations for help, and in 1985 he went to the USA, Britain, the Netherlands, Spain, Sweden and Denmark. In 1985 he was elected to the Transvaal Regional Executive of the UDF, and took over Lekota’s work when he was detained. In 1986 he was again detained from January to March.
In 1994 Morobe became the Chairperson and CEO of the Financial and Fiscal Commission in South Africa. He was part of the Council on Higher Education (CHE), which was established as part of the Higher Education Act of South Africa. He left this post in 1999. He is also the current Chairman of the South African National Parks Board (SANP) and the International Fundraising Consortium, an organisation that provides grants to the non-governmental sector. Morobe has also been appointed as Director on the board of Old Mutual South Africa, and has filled the role of Chairman of Ernst & Young South Africa.