Why does Dropbox require Accessibility?
July 18, 2019 (update)

Found this related post detailing the behavior and another one from the same author explaining how Dropbox deceives the user into getting accessibility permissions.¹

¹These are both close to two years old, but only found them today.

January 18, 2019 (update)

I installed Dropbox today on Mojave and in settings there is something regarding accessibility I had not seen before. Image follows.



From Dropbox help center:


  Why does Dropbox want to turn on accessibility?
  
  Accessibility is an Apple setting that allows applications on your Mac to interact with other applications. For example, it allows the Dropbox badge to work in non-Dropbox applications. 
  
  If you don’t turn on accessibility, you won’t get access to Dropbox features like:
  
  
  The Dropbox badge
  Seeing who’s viewed your files 
  Better prioritization for file syncing
  


Additional information from the Dropbox help center:


How do I turn the Dropbox badge on or off?, and
What is the Dropbox badge?.




July 12, 2016 (old answer)

I found an answer in the community forum of Dropbox. A user sent a ticket to Dropbox asking about this and he posted the reply there. According to the reply, Dropbox needs to control the computer in order to keep our data safe. I'm copying both the ticket and the answer over here.


Ticket #3172481: DB: Dropbox app listed under "Allow the apps below
to control your computer".

I'm sure there's a perfectly reasonable explanation why Dropbox needs
permission to control Mac computers, I'd just quite like to know what it is.

  
  Thank you for writing in regarding the security of your account
  information. I can certainly understand you wanting to ensure the
  security of data you have entrusted to Dropbox. I will be happy to
  explain the measures taken by Dropbox to ensure the security of
  everyone's account.
  
  Permission to control the computer is just another way of saying that
  there are certain system permissions Dropbox needs to function, and
  many of those permissions are to establish secure connections in order
  to protect the data contained within your Dropbox folders.
  
  Operating system permissions prevent files from being viewed or edited
  by unauthorized logins. Permissions can be set any number of ways and
  can be restored manually through a fairly simple process.
  
  Another possible cause of issues with Dropbox arise when conflicts
  exist between your local network sharing or folder redirection
  preferences setup in folders or files within your Dropbox. Similar
  effects can also happen when your Dropbox folder or its system files
  are located in a mounted network drive or a remote location (like
  roaming profiles) since this could cause Dropbox to not have constant
  access or permissions to operate in those locations. This is
  especially an issue if other people in the network could be accessing
  the same data. For these reasons, I'd suggest that you ensure that
  your setup does not include any of the above scenarios.
  
  Additional permissions my be required when syncing mobile devices,
  particularly when photos or photo albums are involved. Since there is
  a great deal of personally identifying information stored in the
  metadata for a photo file (yes, people can find you if they know how
  to decrypt this metadata), some of the additional permissions required
  for our Carousel app, for example, include:
  
  iOS
  
  -Contacts
  -Photos
  -Camera
  -Notifications
  -Background App Refresh
  -Use Cellular Data
  
  Android
  
  -In-app purchases
  -Device &amp; app history
  -Identity
  -Contacts/Calendar
  -Photos/Media/Files
  -Camera/Microphone
  -Wi-Fi connection information
  -Device ID &amp; call information
  
  Facebook does not offer a large amount of granularity or retroactive
  permissions control, so we need to request all the permissions we need
  or even may need in the future.
  
  I want to reiterate the point that all these steps are taken in order
  to prevent access to your data, not to facilitate access to your data.
  Dropbox takes great pride in being a company worth of the trust of our
  users, and we would never ask you to allow us access with the intent
  of violating that trust.
  
  I hope this helps clarify our account security measures regarding
  permissions. Thank you so much for using Dropbox, and please let me
  know if there's anything else I can help with.




Another user asked the following pertinent questions, but he never got any answers for those at least not in that thread.


  "(...) why the need to give up access across the board as opposed to
  an as needed basis? (...) It's hard for me to grasp why giving up OS
  level access to data that Dropbox doesn't need for our use, makes our
  data safer? (...) Syncing data for example seems to work fine without
  the permissions checked. What am I missing? I'm not trying to be
  combative; just hoping to get a better understanding. (...)"