If the answer is no, then I'd go after B&N, Amazon, etc., to stop selling ST fanfiction through their eBook services. If fans want to make fan fiction, they should be the ones who have to distribute it, not mainstream booksellers.
Are they going to stop professional printing companies with offset printers, and public libraries with photocopiers, from printing fanzines that fans bring in?
The legal answer is that they could if they got wind of it.
In reality -- no, of course they won't. But a major online realtor selling copies of fanfic online is a much bigger potential threat to their trademarks and a bigger violation of their copyright, so it would be more rational to go after that than what you just described.