How do I move a single folder from one Subversion repository to another repository?

If you have access the repository itself (not a working copy), you should be able to dump the current repository, filter it to only include information about the docs folder, and load it into the other repository.

Would be something like this:

svnadmin dump /svn/old_repos > ./repository.dump
svndumpfilter include path/to/docs --drop-empty-revs --renumber-revs --preserve-revprops < ./repository.dump > ./docs_only.dump
svnadmin load /svn/new_repos < ./docs_only.dump

Without access to the repository, you cannot maintain the revision history and you have to settle for copying the files into the new repository and committing.

svndumpfilter has a serious flaw - if a file or path was copied from a path you're filtering out to one you're filtering in, svndumpfilter won't be able to fill out the history and the job will fail.

You can use svndumpfilter2 if you experience this problem.