As said, sub hunting in GB is THE duty of a DD. A DD with a sonarman and HHs is the way to kill subs and many a BB will be happy you are around. Poping some smoke to cover a BB from incoming salvos from time to time will actually get you "thanks" from BB players who normally won't talk to anyone. A BB driver likes to have an ASW shp around and some BB players will actually ask you to team with them. (Always do that and never abandon your team mate BB-do your job-cover him and stay out of his way.)
Don't forget that CVs are primary SS targets. So on ASW patrol, you may have to run to the rescue of a CV. They are VERY grateful when you kill a sub. A CVs job is not running from an SS but handling their planes.
In addition to sub hunting, cover can be provided against raiding DDs hoping to launch torpedos into the main battle line. Better that small ships deal with small ships so that your teams BB's can deal with enemy BBs.
In later stages of winning games, you can carefully scout ahead of the slower advancing BB's to give them some vision. Of course, this is very dangerous as the retreating BBs may see only the DD and target it hoping to blind the enemy. Then again, the enemy BB may be so busy dodging the incoming salvos that he has no time to deal with a pesky DD. The DD then becomes very valuable as the scout. And from time to time, you will run down and sink an enemy CV or even a very crippled BB.
In GB, an ASW DD is a very important asset to the team.