Choosing a strong password to protect your online accounts is a good idea, but the best way to keep your data secure is to enable two-factor authentication (sometimes called two-step verification) on the accounts that support it. This usually requires you to enter a code sent to you by SMS or generated by an app, and this has long been true of Dropbox's two-factor system. Now, Dropbox is making it easier to access your account securely by adding mobile prompts for two-factor.
This is similar to something Google did last year—when logging into your account, your phone can pop up a simple confirmation dialog. If you turn it on, Google uses mobile prompts as your default verification mechanism. With Dropbox's solution, the mobile authentication option is treated as a backup rather than the main method of verification. You can get a mobile prompt by clicking "Having trouble getting a code?" after entering your username and password.
You need to be logged into Dropbox on your mobile device for this to work, obviously. After requesting mobile verification, a notification appears on the phone for you to manually confirm that you are the person trying to log into the account. The feature is live now on both iOS and Android.