With Boro fighting against the drop, Southgate said: "What we have to do is accept that there will be plenty of criticism.
"The fans will rightly be disappointed, angry, frustrated. We are most of those things and we have to respond.
"I said last week the ground [atmosphere] was flat and got slaughtered for it, but I thought it was flat.
"But I am the first to stand here and say we got a full house here and I was desperate to put on a performance that would bring them back - and we didn't do that.
"Whatever criticism comes my way, I have got to accept; whatever criticism comes the team's way, we have to accept.
"But we have got to now stay strong together as a dressing room because it will all come at us from outside.
"We have 10 games left in the league, important games, games that can either have a positive finish to the season or one that drags us through a roller-coaster again.
"I have lost football matches before and I am still here to tell the tale.
"But our response now is very, very important. We have got to get pride back, we have got to get the respect of our supporters back and we have to deal with whatever is thrown at us and stay strong."