Wednesday's 2-1 victory over 10-man Hull City did not spare West Ham the jeers of a large number of home supporters.
Allardyce's response on the night was defiant as he cupped an ear to the stands and later said he could not understand the fans' behaviour.
And despite newspaper speculation suggesting West Ham might consider looking for a new manager this summer, Allardyce retains the support of at least one board member.
Brady wrote in the Sun: "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.
"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."
Fans' favourite Mark Noble has appealed to supporters to get behind the team on the run-in as West Ham aim for the 38-point total Allardyce believes will be enough for survival.
The Hull result left them just four points short of that target ahead of Monday night's trip to third-bottom Sunderland.
Watch Sunderland v West Ham live on Sky Sports 1 on Monday night. Coverage begins at 7pm.