To resolve this issue, I am going to provide you with steps to reset the Dropbox application settings without affecting the files in your Dropbox folder.
Just be sure to take note or a screenshot of any Selective Sync settings you may have applied prior to re-installing as you will need to re-apply these settings after this process.
First, please save and quit all programs accessing files in the Dropbox folder, and then follow these instructions:
1) Open your Terminal app (Located at /Applications/Utilities/Terminal)
2) Copy and paste the following line (highlighted below) into the Terminal and press ENTER. PLEASE make sure you copy and paste this command (don't type it by hand), as getting it wrong could cause some harm.
You'll be prompted for your computer's admin password (not your Dropbox password) after entering the first command. Keep in mind that the password field in the terminal will remain blank as you type your password. After you type it, just press ENTER. You'll know that these instructions complete normally once the initial line of text (called prompt) comes up again:
sudo mv ~/.dropbox ~/.Trash/dropbox.old
Please note that once the prompt displays, the instruction completed the requested task. If the command returns an error, or does not respond after an hour, you may want to try restarting your computer and trying this step again.
3) Restart Dropbox from the Application folder, then please sign in and apply any Selective Sync settings you may have had prior to this reset during the sign in process. Your account will take a few moments to reindex the files and sync any pending changes.
If Dropbox still doesn't start then please copy and send me the result from the Terminal commands and any error messages. Alternatively, if it's easier you can also take a screenshot and attach it to your next reply here.
The 'sudo' will ask for your password (local account password, not Dropbox one). Next, run the Dropbox application again. The cache will be rebuild. Take care, don't change these settings/permissions anymore.