I applied through their website, which was pretty much just a resume submission. About a week later, I received an email asking me to fill out an online questionnaire. The first three questions were sample customer service emails that I needed to answer as a member of the Dropbox Support team. The other questions asked me to explain how Dropbox changed the way I use computers, explain the most interesting thing I've learned recently, and a wildcard question asking me to explain a world where "Dropbox is pre-installed on your toaster."
A week later, I was asked to do a phone interview. I spoke to a member of the User Ops team, and the next day I received an email saying they wanted to schedule a phone interview with an additional member of the User Ops team. I don't quite remember all of the questions from both interviews, but the first one mostly contained questions similar to the questionnaire about certain hypothetical troubleshooting situations and how I should respond. The second interview focused more on what type of information Dropbox could learn about a user based on their activity.