The South African world number 25 told BBC World Service Sport that he had received threats from people who had bet on his match against Denis Istomin.
"It pretty much happens after every match regardless of the circumstances," the 30-year-old said.
"People who have bet on the match tend to take to social media and say very inappropriate things."
Anderson, who reached the fourth round of Wimbledon last year before losing to eventual champion Novak Djokovic, said abuse on social media was commonplace.
"It's just something you have to deal with as a professional athlete who is in the limelight," added Anderson, who lost in five sets to Uzbekistan's Istomin.
"You can't obviously read too much into it, you let it run off your back. It would be nicer if you didn't [have to], but that's the reality of the world we live in."
Kevin Anderson has revealed he received "death threats" after his first-round loss at Wimbledon on Monday.