In an update on its website, Transnet Port Terminals said the strike had impacted waterside and landside operations at its Durban port, which handles 65% of South Africa's container volumes.
The strike could also disrupt fruit exports from Transnet's Cape Town port just as the deciduous fruit season begins.
The Transnet strike is also set to worsen the mining industry's logistics woes.
Even before the strike, South Africa's Minerals Council had projected a revenue loss of 50 billion rand ($2.76 billion) this year, compared to 35 billion rand in 2021, as Transnet's underperformance throttles exports.
Miners Thungela Resources, Kumba Iron Ore and Jupiter Mines have warned that the strike is likely to impact coal, iron ore and manganese production and exports.
Transnet has said it will meet union leaders on Wednesday to continue wage negotiations.