Inspiring other creative people is one of the huge perks of this job. The influence your work has on other writers forms connections and establishes a sense of continuity. If there is such a thing as reincarnation, that's how it works: the wisdom and craftsmanship belonging to generations of storytellers pasts is reborn in the souls of the writers who came after them.
I think book writers mainly inspire other book writers, but we also have an impact on artists who work in other mediums, like poets, musicians and movie makers. For me there is no finer accolade than learning that a poet has written a verse about my characters, a rock band has named themselves after one of my story elements, or that a reader has put together a video dream cast for one of my series.
The visual arts I've inspired are the most fun for me to discover, especially as I'm so often influenced by visual artists. A few days ago a friend who hangs out on Deviant Art sent me a link to this StarDoc-inspired piece by Autumn Haynes (.jpg of painting posted here with the artist's permission.) Seeing my characters through another artist's eyes is always delightful, but it also recharges my creative batteries, so that when I go to write new characters and stories I really think about it and put everything I've got into it.
I like to contact artists and express my appreciation for their efforts, but it's not always possible. Some choose not to respond to e-mail (probably because hearing directly from an author either scares or annoys them), and then sometimes there is just no way to reach them. My publishers send me reader mail that includes art like this portrait of Jayr and Byrne from the Darkyn series. Unfortunately in this case the publisher chose to send me just the contents of the letter to save on postage; they discarded the envelope it was sent to them in. Which means I have no return address. I did try searching for the artist online but I never found any contact info. If you're out there, Corine, thank you for this.
What should you do with reader art? It's really up to you and how much value you place on it. I think it's nice to collect and preserve it as part of your personal writing history. I've framed a lot of mine and put it up on my Wall of Why, and eventually I plan to put together a photo book of my collection for my kids. Other ideas: start a scrapbook, make a wall collage or photoshop them into a poster for your writing space.
If you're a reader who creates novel-inspired art, definitely try to send a copy to the author. I can't promise that you'll always get a reply or a thank-you in return, but you may end up giving a creative boost to the one whose work inspired you.