Ms Graham said the oystercatcher parents needed to be given space to feed and raise their young without disturbance, and the roped-off area was there for a good reason.
"These precious little balls of fur are totally defenceless against a dog, and for the next few weeks they're particularly vulnerable. I'm pleading with people to leave their dogs at home and to not to disturb this little family," she said.
Ms Graham said it was important that people showed respect and kept their distance from the nesting parents, nicknamed Harry and Sophie, and their offspring.
Chicks fly when they are between six and seven weeks old.
Ms Graham said as the chicks started to explore their habitat they would become a "moving target" for dogs and other wildlife.
Department of Conversation staff were not available for comment last night.