"We're trying to be cautious and ensure people's safety in the case of a further aftershock that might dislodge this thing,''
Mr Brown said engineers had started installing the first of a number of restraining brackets on the side of the lift shaft this afternoon. More work would be carried out tomorrow.
The restraints, devised by seismic engineers, were designed to hold the lift shaft to the building in an aftershock.
Council spokesman Richard MacLean earlier told RNZ there were "quite a few'' apartments, organisations and businesses in the area.
The street closure would disrupt a large number of people for a few days, he said.
Meanwhile, Mr Brown said assessments were continuing on other buildings damaged in the quake.
"Building owners, I'd have to say have, been extraordinarily responsible, and engineers have had a really busy few days with reinforcements called in from outside of Wellington to help get through the volume of work that is required.''
The city was shaken by a magnitude 4.2 earthquake today, but GNS Science has said it was unrelated to Sunday's magnitude 6.5 quake.
The quake was located 40km northwest of Wellington at a depth of 68km.
GeoNet received over 1300 reports of people feeling the quake.
There have been over 1500 aftershocks since Sunday's shake near Seddon.