At 1555 GMT, the rand was down 0.6% at 14.5125 against the dollar, extending last week's losses over riots and looting that erupted in the wake of former President Jacob Zuma's jailing.
The rand is highly susceptible to swings in sentiment on global markets and prices of metals like gold and platinum which are major South African exports.
The dollar and other safe-haven assets climbed, after U.S. officials said the more contagious Delta variant first detected in India was now the dominant coronavirus strain worldwide.
On the Johannesburg Stock Exchange, the Top-40 Index dropped 2.7% to 58,795 points and the broader All-Share index fell 2.6% to 64,805 points.
Precious metals miners like Sibanye Stillwater, Impala Platinum and Northam Platinum were among the biggest losers on the blue-chip index, down 6.5%, 4.7% and 4.5% respectively.
Most firms on the blue-chip index were down on Monday, however, with finance companies especially still suffering from the blow to the rand and economy from last week's unrest.
(Reporting by Emma Rumney and Alexander Winning; Editing by Lisa Shumaker)