At the beach, were normally told to swim between the flags. Rockdale Council wanted swimmers swimming in between the nets (netted bathing pools) as a means of keeping people safe in the water.
Source: Poulus, J. 1986. "Swimmers risk death: Mayors' warning on shark attack threat." The St George and Sutherland Shire Leader, January 16: 1.
Even in the 1970s, the council was concerned. It even produced signage to alert swimmers in four languages (English, Italian, German and Macedonian). The calmer waters have always been popular with those from immigrant backgrounds, even to this day. In 1986, Rockdale Council believed that 80 % of swimmers at Lady Robinsons Beach were of immigrant background.
Source: Anonymous. 1972. "Shark Warning In Four Languages." The St George and Sutherland Shire Leader, September 20:11.
For the record, the last fatal shark attack in the Georges River/Botany Bay catchment was in 1946 at Oatley Bay in Oatley when Valma Tegel (14) died after having her left leg ripped off by a shark in water's just one metre deep. The last shark attack in the bay itself was in February 1940 when John William Eke sustained injuries to his arms at North Brighton (Kyeemagh). There have been no fatal deaths from a shark attack in the bay itself.
The risk of shark attack is still there but doesn't seem to be as of much concern as now. These days you are more likely to drown in Botany Bay due to the structure of the beach itself. A five-year boy drowned at Dolls Point in late 2013. The water depth increased sharply not too far offshore and people may not be able to see it. Currents in the bay have also played their part as well.
Updated November 2018 - Clearer image added.