Police were called after the CCTV operatives saw the males attempting to scale the city centre Church of England building.
A council report said it was feared the climbers may have been trying to harm peregrine falcons, which traditionally nest on the cathedral in springtime.
However, officers attended the incident and concluded this was not the case, and the males were told to leave the area.
The report, written by the council's portfolio holder for communities, Maureen Cummings, said: "Operatives feared the worst when they picked up two males attempting to climb Wakefield Cathedral.
"The operatives were initially concerned the climbers could have been after the Peregrine Falcons that nest at this time of year.
"Police were called and two males were told to get down. It turned out the males were drunk and had started to climb for fun.
"Officers gave them some good advice on the dangers they faced and sent them on their way."
The report did not say when the incident had taken place.
Local Democracy Reporting Service