England will report Montenegro for racial abuse following their 5-1 in Podgorica.
"It is unacceptable. I definitely heard abuse of Danny Rose when he got booked. There's no doubt in my mind it happened. I know what I heard. We have to make sure our players feel supported, they know the dressing room is there and we as a group of staff are there for them," said Southgate.
Raheem Sterling and Callum Hudson-Odoi were also subject to abuse.
"There should be a real punishment for this, not just the two or three people who were doing it - it needs to be a collective thing. This place holds 15,000. The punishment should be, whatever nation it is, if your fans are chanting racist abuse then it should be the whole stadium so no-one can come and watch. When the ban is lifted, the fans will think twice. They all love football, they all want to come and watch their nation so it will make them think twice before doing something silly like that," said Sterling.