How Not to Do It 101
However. I do respect the right of any artist to say, "Listen, can you just appreciate what I produce and not play in the same sandbox? Thanks for understanding." Not everyone is comfortable with fans producing related works, no matter what the nature of those works are, and they have absolutely no obligation to be comfortable with it. A lot of fansites are quite respectful of this and will tell people posting fanfiction in one fandom, "You can post crossovers as long as they aren't crossovers with any Anne Rice work," for instance, to avoid any legal confrontations with Rice's lawyers over derivative works.
Another however. If you are an artist who would really rather not have your fans posting fanfiction taking place in your universe(s), all you need to say to get that across is the following:
"I would like to ask that no one make available for public consumption any fanfiction based on my characters or taking place in my world. I understand that this is probably something some people would like to do, but I simply don't feel comfortable with it, for a number of reasons I'd really rather not get into. I could change my mind in the future but this is the way I feel right now. I appreciate everyone respecting my feelings about this. Thank you."
(Anyone who wants may feel free to copy and paste or edit the above to suit your purposes.)
Yet another however. The reason for making a statement like the above is that, despite many writers who are somewhat successful having a problem with fanfiction, I have yet to come across one with this stance who has managed to say something like the above, instead choosing to enumerate the many, many reasons why he or she is anti-fanfiction and in the process alienating their fans; pre-emptively preventing many potential fans from deciding to even try their work; mischaracterizing the nature of fanfiction in general and the works written by their fans specifically; disregarding any possible good to come from fanworks at any time or place; demonizing anyone who enjoys reading the fanworks; demonizing anyone who benefits from fanworks (beneficiaries of fundraisers, for instance); using horrifically bad analogies to communicate how the artist feels about being "violated" this way that inevitably offend rape victims, allies of rape victims and/or parents (comparing the work to human offspring), at best; and creating the impression that he or she is a know-nothing, selfish, big-headed poopy-brained small child who is dissing a lot of people in the process of demonstrating what a Speshul Flower he or she is.
This seldom goes over even a LITTLE well.
There will be a few other anti-fanfiction people who will respond, "Well, good god, who could blame her?" Others will point out the legalities. Yes, we're all familiar with the fact that a lot of fanfiction is dreck, that the author owns the intellectual property. This isn't about the law; it's about perception.
When an artist is perceived to be generous and encouraging to fans despite having the legal ability to sue them into the next century over something that absolutely does NOT detract from the artist's ability to make money from their own work, that is not taken lightly. Fans will appreciate that more than most authors realize.
Conversely, when an artist is throwing a hissy fit about fanworks that don't involve outright theft of a large portion of the artist's output, selling the author's own work and making money from it that does not go to the author (mind you, anyone selling books second-hand is doing this completely legally) or characterizing large portions of a dedicated fandom as hack writers, rapists (of fictional characters) and/or pornographers (and not in a good way) the opposite of good-will is being sown in that fandom. The author can effectively sour a large portion of a dedicated and active fan base, potentially driving them away from her universe and anything else she plans to write for the rest of what is left of her career.
It's about PR, people. There are simply some things you don't ever want to be caught saying in public if you want to continue to sell your work, period. No one is forcing anyone to like having fanfiction written based on their work. (As someone who's had fanfiction written based on my fanfiction I can say that it's a good thing no one is forcing me to like it.) But if you're going to take the step of looking at it as a Bad Thing on principle, rather than something that can have an awful lot of benefits, at least don't dig yourself a hole by 'splaining, in graphic detail, all of the problems you have with it. Be brief and to the point and don't get into the nitty-gritty or paint everyone who does this thing you detest with a very, very broad and nasty brush that will come back to haunt you later.
And always, always, emphasize that this is just how you feel right now and you could change your mind at any time. Leave the door open to that generosity of spirit. Don't burn bridges. You'll be glad in the long-run.