Can I host my published software at Dropbox?
May I keep the files for online availability at Dropbox as public link.


Unless you agreed to some restrictive copyright agreement regarding that software or your employer imposes some restrictions (both of which I would consider odd, but then I do not know the customs of your field), it is your software and you can do with it whatever you want.


  If that's not a good idea what are the options.



If you did not already do so, you can see whether the journal allows you to publish software as a supplemental material or similar for your publication. Even if they do not state so on their site, they may still do it on request.
In addition, I would suggest that you publish your software on Github, Sourceforge or some other repository designed for publishing software. This way, you can easily update the software, others can contribute to your software and so on. If you do this, I recommend to add a link to the repository in your paper (or the source code attached to your paper), so people can obtain the latest version.

I am no expert on Dropbox, but it does not seem as long-lasting to me as most dedicated software repositories and probably has no special features for hosting software.