The Cottagers are currently four points adrift of safety at the bottom of the table following four straight defeats.
Their troubles were increased on Tuesday night with a shock 1-0 loss at home to Sheffield United in an FA Cup fourth-round replay.
"The most important thing is that we don't lose the togetherness because that is the only way we're going to get out of this mess," Meulensteen told the club's official website.
"If you start falling apart, then you have an even bigger problem and that is what cannot happen.
"The most important message to the fans is to not add to the problems we already have, but to be part of the solution and make sure you keep sticking together, keep supporting the team and players most importantly."
Fulham are back in action on Sunday when they make the trip to champions Manchester United in the Premier League.