Councillor Kirstie Hurst-Knight is asking residents to be careful along the riverbank and be considerate of those whose homes neighbour parks and other open spaces.
The town council says it has also received numerous reports of late night disturbances and small scale fires.
Councillor Hurst-Knight said: "I appreciate that the opportunity to have a bit more freedom, as a result of the changes to lockdown, is being taken by many. Who can blame them?
"We have been asked to stay at home for a long time up until now and the weather has been wonderful.
"I would however ask people to not be reckless with their own or others' safety.
"Swimming in the river can be dangerous, even with the apparent low water levels.
"Our emergency services are stretched at the moment and if we can reduce the need for them to attend avoidable situations then that can only be a good thing."
The mayor said people are also advised against lighting of barbecues and fires in public areas.
"I note that there have been a number of reports of anti-social behaviour in and around the town in recent days," she added.
"While there is a need for people to let some steam off and take advantage of the opportunities to meet others in public and open spaces, it is not fair that those whose homes are close to such places suffer from disturbances late on and into the evening.
"In these difficult times when so many of us have foregone our usual way of life can I ask that we all try and think about our actions and how they impact others and, on the other hand, that we all show a little tolerance to those who might unintentionally provoke us."
Deputy town mayor Councillor Sarah Barlow has also urged people to be safe and considerate.
She said: "The river poses a threat all year round. There is still an undercurrent that can put you in difficulty and the river is still full of debris from the flooding earlier in the year.
"I know that many people are using our parks and open spaces. I am glad that we live in a town where such opportunities are readily available but I am concerned that many people are leaving rubbish behind for either others to clear up or for it to end up in the river or hedgerows.
"With regard to anti-social behaviour, I am grateful that the town council has an effective dialogue with the local police so that we can share with them our concerns and incidents reported to us.
"As a council we are working with the emergency services and local schools to help provide a safer community for all."