How can I change the location which Anki uses to save files?

You may be able to use Symbolic Links to create a link between where Anki stores its files and a folder within DropBox. Information on how to create symbolic links and how they differ from Aliases can be found here:

http://hints.macworld.com/article.php?story=2001110610290643

When going to documents folder there is a readme with link to the user manual. That page has several ways to pick the data location. Portable mode is my pick. Standalone also looks good.

Adding the -b flag was not working until I read

On Windows, right-click on the shortcut, choose Properties, select the Shortcut tab, and add "-b \path\to\data\folder" after the path to the program…

which should leave you with something like

C:\Program Files\Anki\anki.exe" -b "C:\AnkiDataFolder"

There are options for Macintosh and flash drive too, all on that same page.

As of Anki 2.1, the ANKI_BASE environment variable should be work per this. However, you might need to set it in a Launch Daemon / Agent for Anki to figure it out if you use Spotlight. Tested 2.1.0beta4.