What are the risks in backing up private keys on services like Dropbox?

There's also the whole "Dropbox can read your stuff" problem.

What you should do is encrypt everything before putting it into Dropbox. Use something like KeePass as a secrets vault. Put a good password on it. KeePass will encrypt locally, before putting your stuff into Dropbox. You will then use KeePass on other computers to access those secrets.

Take a look at:

https://superuser.com/questions/351525/is-keeping-a-keepass-file-in-dropbox-safe

So, in summary, encrypt locally. Use Dropbox to sync those encrypted files.