Alpine ski World Cup organisers provided bodyguards for Maze, 24 hours after she beat the record of points scored in the overall World Cup thanks to her victory in a downhill in the German resort.
"It's a shame but things like that happen," the head of the organising committee, Peter Fischer, told journalists, adding that the anonymous threats had been received at the local ski club.
"What's happening to me is really ugly but I suppose it's part of the game. I don't have words to express what I feel," said Maze, who finished fourth, 0.43 behind Fenninger.
"At the start, I tried not to think about it and to focus on my race only. It's a new situation for me to have to face things like that and I don't know how I should react."
The Slovenian's fourth place took her overall points tally to 2,074 with seven races remaining. On Saturday, she became the first racer to surpass 2,000 points in a season.
Former super-combined world champion Fenninger earned her third victory, her first in a speed event. The last Austrian to win a women's super-G was Elisabeth Goergl in the 2009-2010 season.
"It's my first victory in a super-G and I'm very proud of it as I feel more like a giant slalom specialist. But I want to be strong in every discipline," Fenninger said.
Germany's Maria Hoefl-Riesch missed out on victory in her home resort by 0.20 seconds while American Julia Mancuso was third, a further 0.03 seconds behind.