They will require a sponsor and escort for events such as press conferences and interviews.
Speaking to Radio New Zealand's MediaWatch programme, Dr Smith said it was not acceptable for members of the media to ignore Parliament's Standing Orders, regardless of whether it was in the public interest.
"What was troubling was the impression created that the Herald would decide whether the Standing Orders should be breached or not - that it was a decision for them to make - and that is not acceptable.
"The Standing Orders in Parliament are not just general protocols - they are actually the rules of Parliament and all members of ... the press gallery have to comply with them.
"If it was allowed, there would likely be an escalation of incidents and we'd have no choice but to block off the gallery and I think that would be a real shame.''
Dr Smith said the sanction was "not a huge handbreak'' for the newspaper, despite it being just six weeks out from the general election.
"It's just that they won't have the convenience of moving around the precinct freely.
"Now, with Parliament not sitting, that's not the biggest inconvenience in the world either and I think this cry over the disproportionate nature of the sanction is perhaps a wee bit over the top.''
Asked if he or his National Party colleagues were worried that he had angered the biggest-selling daily paper in the country, Dr Smith said: "I hope that if they're angry they're angry with me as the Speaker and not with the National Party because the Speaker's decisions have nothing to do with the National Party whatsoever.''
The Herald's editor Tim Murphy had said the sanction was "totally out of sync with any sort of proportional response''.
In June last year, the Speaker suspended 22 Parliament car-parking permits held by TV One and TV3 after their crews pursued former MP Chris Carter down stairwells in the complex and filmed in his secretary's office in his absence.
"We decided that this was so news-worthy and so extraordinary... that it justified being shown to the public to show how close MPS came to being at risk, Mr Murphy said.''
However, the accreditation of their journalists and camera operators to work in Parliament was not affected.
At the time, Dr Smith said they were serious breaches of the rules, including filming in a prohibited area and entering an MP's office area without permission.
Jane Patterson, chairwoman of the press gallery, said earlier she was surprised the Speaker had decided to suspend the accreditation of all of the Herald office, rather than only the person who took the photo.
"While I understand why the Speaker felt the need to act in response to a breach of the rules, I do think the sanction is severe."