More than 30 years ago I received a pair of life-changing books that I read and annotated to pieces so I had to buy fresh copies, that I still read and re-read and use with pleasure, and finally a few weeks ago I emailed the author the fan letter. This was a cookbook author. (Cookbooks are the one kind of book I would never want as e-books only. I love love love them, and especially those by this author.)
I told the author how much the books have meant to me, what recipes I liked and how they helped sustain me and friends in good times and in lean times. I did not expect a reply, but she wrote that I made her day, and so on. If you have ever received an appreciative letter about your work, you know how it refreshes you. This was the thought finally drove me to take a minute and send my fellow writer, the cookbook author, my long-overdue verbal bouquet. I recommend this. It feels like a deposit in the Bank of Good Karma.
What I don't recommend is trying to establish a correspondence. A real writer is going to be too busy writing to be your penpal. (OMG, who remembers "penpals"??) I sent then-Vanity Fair columnist Christopher Hitchens my article about a TV show on which he had written another opinion. He wrote me back to say he appreciated my point of view. (I was amazed he had received, opened and read my letter at all.) I wrote him back but did not hear from him again. This wasn't rude. He's a writer.