The man was aged in his 50s or 60s and "very drunk", he said.
After the establishment, which Walker declined to name, ejected the other party, the man shouted an objectionable comment directed at him and the shirt he was wearing that showed his support for Labour.
"People wearing a Labour T-shirt should be shot in the head," the man allegedly yelled.
Last night Walker called the behaviour "the sort of 'Trumpian' reaction" New Zealand would be better off without.
He said negative conduct surrounding politics was on the increase, to his mind, for a number of reasons.
National's Southland candidate Joseph Mooney said overnight on Saturday National Party election hoardings in Centennial Ave at the western end of Alexandra were toppled.
No other party's hoardings were damaged in the street, he said.
Mooney, a Queenstown-based lawyer, was named on July 31 to replace Hamish Walker as the candidate for the new Southland electorate.