Councillor Liz Colling said: "I appreciate this is a very difficult time for everyone but we must all take our collective responsibility to protect each other seriously and adhere to the strict Government guidance on social distancing and the closure of bars, restaurants and cafés.
"Please stop and think about the very serious nature of the ongoing situation and how vital it is that we all follow the official guidance.
"It is not worth putting yourself and those around you at risk."
People flocked to Scarborough, Filey and Whitby yesterday (Saturday March 21) despite government advice everyone should be social distancing.
There were reports of long queues outside cafes on seafront areas as people made the most of the good weather.
The Government has ordered all restaurants and cafes not to allow people to sit inside but have said they can operate takeaway services.
Cllr Colling continued: "I am therefore urging all residents, visitors and businesses to act responsibly and compassionately with immediate effect by following what the Government is asking us to do. It is incumbent on all of us to work together to try to prevent the spread of coronavirus.
"I hope I can count on everyone’s support to do what is right for our borough and our country at this very challenging time.”
John Senior, chairman of the South Bay Traders Association Scarborough (SBTA), joined in the plea for people to stay at home and businesses to close after speaking with Mike Greene, CEO of Scarborough Council.
He said: "Mike Greene has asked that all non-essential businesses close with immediate effect.
"As Chairman of the SBTA Scarborough I am asking all of our members to close immediately in order to remove the temptation for unnecessary gatherings and help support the fight against Covid-19.
"Don't forget that once this damned thing is beaten our beautiful North Yorkshire Coast will still be here for you to enjoy and we all look forward to welcoming you back as soon as is safe, in the meantime please stay safely at home!"
Several Facebook users shared their fears that so many people visiting the towns would encourage the spread of coronavirus.
One posted: "I'm sick of it, honestly. People round here abiding to the rules, then visitors are cramming the town at weekend - it's sheer madness and spoiling for the ones are doing all they can."
And one said: "It is people like this that are going to cause the Government to eventually just put a total lock down in place and restrict all movements. Selfish."
In a press conference on Friday the Government announced that pubs, restaurants, cafes, gyms and leisure centres should close as infections spread easily in closed spaces when people gather together.
Boris Johnson urged everyone to take steps to reduce unnecessary social interaction.
Although people are encourage to go for a walk or exercise outdoors, they should do so by staying more than two metres from others.
There is also advice to avoid gatherings with friends and family, and instead keep in touch using remote technology such as phone, internet, and social media.