The 83-year-old, who made the hit series in a studio on Slough Trading Estate in the 1960s and suffered from mixed dementia, died in his sleep on Wednesday afternoon.
Jamie Anderson said: "I just wanted to thank everyone for their incredibly kind messages of support and for sharing their happy childhood memories - inspired by Dad's work.
"I know Dad would have been blown away by the support, positivity, and kind words.
"I think the saddest thing would have been if he had passed without being noticed, but the response has been the total opposite. Thank you."
The producer, who also worked on Captain Scarlet and Joe 90, was diagnosed with mixed dementia in 2010 and moved to a care home in October after his condition worsened.
The father and son duo helped raise awareness for the Alzheimer's Society and Jamie took part in the Windsor Memory Walk at Dorney Lake on October to raise funds for the charity.
To make a donation to the Society in memory of Mr Anderson, visit www.alzheimers.org.uk
Mr Anderson is survived by three children from previous marriages, Joy, Linda and