The 73-year-old employee was so shaken by the threats, he tendered his resignation, but was later talked round after colleagues rallied behind him.
The railway's director, Malvern Tipping, wrote to the local police sergeant to call for action after multiple incidents of staff being victims of rude and aggressive behaviour by people refusing to follow the rules.
Officers have now visited the railway to take details of the incident, and urged staff to dial 999 in the incidence of further abuse.
Malvern said: "Although we would have appreciated an earlier response in this respect, my staff and I are very grateful for the support and back-up which the police have offered us with respect to abuse and attacks from individuals refusing to wear face masks.
"The police have inspected our Covid-19 signage and pronounced their satisfaction with it.
"Moreover, the police have told us to dial 999 in the event of any more abusive incidents, whether physical or otherwise, against cliff railway staff.
"Again, we are most grateful for that and will be working with the police as and when situations arise."