While they can mean similar things in certain contexts, they aren't quite used interchangeably.
I'd say that the word "deteriorate" is more common, and is a verb that can be used to refer to worsening condition of something.
While the word "degenerate" can technically be used as a verb as well, it's much more commonly used as an adjective or noun referring to something that is morally corrupt. In my own personal experience I have rarely heard it used as a verb.