The Proteas, ranked number one in the world, were thrashed by 241 runs in the first Test in Durban, with the second Test in Cape Town starting on Saturday.
Captain Hashim Amla is in poor form, batsman AB de Villiers is struggling with his workload and bowler Dale Steyn has a shoulder injury.
"There is a lot that went wrong in this game," said Domingo.
"There are a lot of questions about the batting line-up, about AB's retirement and Dale's injury."
Domingo said it was important for his players to remain a "tight unit".
"There will be a lot of comments on Twitter and Facebook, but we have to keep believing in what the team is capable of," he added.
Domingo gave further momentum to speculation De Villiers will soon start missing matches for his country - including some Tests.
De Villiers, 31, plays all three formats of the game for South Africa and also features in the Indian Premier League.
"We all want AB to play until he is 50, but it is not going to happen," said Domingo.
"It is always a fine line. You want a full team, but you have to manage your best players so they play for a long period of time. We are always grappling with it."
South Africa have made two additions to their squad for the second Test, bringing in wicketkeeper-batsman Quinton de Kock, 23, and uncapped pace bowler Chris Morris, 28.
The former may ease the burden on De Villiers, who reluctantly wore the gloves in his 103rd Test appearance. Morris, meanwhile, joins Kagiso Rabada in providing cover for Steyn, who seems unlikely to play at Newlands.
Captain Amla described his team's poor form with the bat as "a confidence thing", and acknowledged his lack of runs was a concern.
The 32-year-old last made a Test century in December 2014, hitting 208 against West Indies, and only scored 118 runs in seven innings during his side's series defeat in India.
BBC cricket correspondent Jonathan Agnew wrote during the first Test that Amla "offers nothing to this side in terms of leadership", and South African journalist and Test Match Special commentator Neil Manthorp believes Amla should relinquish the captaincy.
"There is absolutely no shame in stepping away from the captaincy. If it means runs, then you're doing it for the team," he tweeted.
Asked if he wanted to continue in the role, Amla said: "I definitely do. I've enjoyed every minute of it. It's just been disappointing not to get the runs.
"As captain you want to lead from the front, so hopefully I can turn it around in the next match."
South Africa squad: Hashim Amla (capt), Dean Elgar, Stiaan van Zyl, AB de Villiers, Faf du Plessis, Temba Bavuma, JP Duminy, Kyle Abbott, Dale Steyn, Dane Piedt, Morne Morkel, Rilee Rossouw, Quinton de Kock (wkt), Kagiso Rabada, Chris Morris.
South Africa are beset by problems going into the second Test against England, says coach Russell Domingo.