Allardyce’s team laboured to a 2-1 victory over Hull on Wednesday night despite playing against 10 men for much of the match following a 23rd-minute red card for City goalkeeper Allan McGregor.
Some Hammers fans voiced their displeasure at the manner of the performance - even though the three points saw West Ham ease further clear of relegation trouble and up to mid-table in the Premier League.
Allardyce cupped his ear to their boos in disbelief and Brady believes her manager was right to be upset.
``Sam had selected a team for a low-scoring match, knowing what to expect against Hull, and could not have predicted he needed instead a team to play against 10 men,`` said Brady writing her weekly diary in The Sun.
"His actions on the pitch were prudent and, at this stage of the season, correct. There is no way I can accept my team being booed. It undermines them when, in particular, they are giving everything they have to make up for the bad time the club endured before Christmas. Big Sam is so upset he complains that he has never heard a winning team booed. While this might not be strictly true, I doubt you would get the same at Fulham or Sunderland if they had won."
Brady’s comments follow Mark Noble’s plea to the West Ham supporters to get behind the team for the remainder of the season.
Noble admitted the display against Hull was not great but said that booing would do nothing to help the situation.
Speaking to the media on Thursday he said:
"I appreciate that we did not play our best football and that, at times, it wasn't pretty.But the anxiety from the stands can spill onto the pitch and get into our minds, as much as we try to ignore it. When that happens, some players can be affected and they make mistakes that they would not normally make.
"For that reason, I would ask that you give your backing to the boys because I can tell you that we are giving 100 per cent for the shirt and 100 per cent for West Ham United.''