Instead of just deleting a branch, it might be a good idea to keep a tag to the old branch.
# Get remote branches verbosely (with SHA and some description)
git branch -a -v
# Pick the old branch that you want to archive and note its SHA
git tag archive/<old-branch-name> <old-branch-SHA>
# Delete the local branch
git branch -d <old-branch-name>
# Push the tag to remote repository
git push --tags
# Delete the remote branch
git push origin :<old-branch-name>
# (Optional for other people monitoring the remote repository)
# This removes outdated branches from their local repository
git remote prune origin
References