Chances are, as this PCWorld article discusses, Dropbox is right in insisting it has not been hacked. According to Dropbox, the passwords were actually taken from other websites or by utilizing other methods.
If true, hackers compromised accounts because the victims reused their passwords across multiple websites. What this occurrence stresses is the importance of practicing good password hygiene, especially since the level of security from website to website can differ significantly.
The solution is to use a different, lengthy and random password on every website. Obviously, this makes your passwords impossible to remember. However, utilizing a password vault can definitely help by generating the strong, random passwords, filling in the forms for you and syncing between your various computers and devices. There is no excuse not to use unique and strong passwords with every website, and you will be much safer if you do.