The 23-year-old Halep, ranked third in the world, had received a death threat via Twitter on Thursday from a Danish fan.
The Stuttgart tournament’s organisers confirmed they have stepped up security around her.
“I can’t comment on it. I am here to play tennis. Other people will take care of it,” said Halep after her 6-4, 6-4 win over Errani in a tight quarter-final on the indoor clay court.
The obsessed fan is believed to have bombarded Halep with a series of messages via social media.
The Romanian showed no signs of the off-court distraction to claim victory which is set to see her take Maria Sharapova’s place as second in the world rankings on Monday.
She finally converted her fourth match point against Errani in a match which featured 16 breaks in the 20 games.
“It was a tough game, she defended very well and it was hard to finish points off against her,” said Halep.
“I’m looking forward to the semi-finals now.”
Halep, the 2014 French Open finalist, now faces former world number one and boxing fan Wozniacki after the Dane floored Spain’s Carla Suarez Navarro 6-0, 6-3 in just over an hour.
“I surprised myself by having fun out there today, I don’t normally like clay, but it was nice to slide and run,” said fourth-seed Wozniacki, who says she sometimes spars in training.
“I like to box from time to time, it’s a great cardio work-out,” added the 24-year-old.
Halep is the highest seeded player still in Stuttgart after reigning champion Sharapova, who had won all of her three previous tournament appearances in Stuttgart, was beaten by Germany’s Angelique Kerber on Thursday in the second round.
Kerber also reached the semi-finals with a straight sets win over sixth-seed Ekaterina Makarova of Russia.
The 27-year-old needed just an hour for a 6-3, 6-2 win to celebrate her ninth win in succession after winning the Charleston tournament a fortnight ago.
“I’m speechless over how amazing the atmosphere is here on the centre court,” beamed Kerber.
“I have rediscovered my joy and passion for tennis and I am enjoying every minute on court,” she said, having broken a poor run of results with her win in South Carolina.
In the semi-finals, the in-form German will play Madison Brengle of the USA, who recovered from losing the first set to beat Caroline Garcia of France 3-6, 6-3, 6-3. (AFP)