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 & app history -Identity -Contacts/Calendar -Photos/Media/Files -Camera/Microphone -Wi-Fi connection information -Device ID & 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. (...)"