Yes, writers do need "calling cards."
Today, no one today feels obliged to call you (whether in person or on the phone), email you, or even visit your website. So why have one?
1) It's puts a name to your face.
2) It tells the recipient that you're a writer (and what type of writer you are: fiction, nonfiction, short stories, books, articles, etc.)
3) It tells how to contact you, just in case the receiver of your cards wants to. You never know!
4) It shows that you're professional, that you take your work seriously.
5) It's easier to exchange information with someone, especially a fellow writer or editor or agent.
What needs to be on your business card?
1) Your name (obviously), as well as any pen names you use.
2) Phone number (cell or home or both -- you can also set up a Google voice mail account that has a separate phone number for free)
3) Email address (You can set up one just for your writing business so that your personal email isn't overwhelmed or your writing business emails get lost amid your personal ones.)
4) Website and/or blog
You can also include:
1) Your photo (This makes it even easier for the receiver of your business card to put your name and face together and remember you.)
2) Your address (However, I recommend getting a post office box instead of listing your home address. P.O. boxes don't cost that much. I've used one for years.)
3) Your books or articles (If you have many, perhaps just list the most recent ones.)
4) Your services (Do you ghost write? Edit or proofread? Research?)
What to do when you get a business card from an editor or agent or fellow writer:
1) Write on the back of the card when and where you met them. It helps to remember who they are. You'll also need that information if or when you decide to contact them so they can also remember who you are.
2) Keep it in a place where you can find it again. Some people use business card albums (like photo albums, but with little slots that fit business cards) to organize and file the cards they receive. You can get these at places like Office Max or Staples. You can keep them in "recipe" boxes. You can staple or tape them in notebooks. The idea is to be able to find them when you need them.
3) Write a short note or send a brief email (or visit their website and leave a comment) not long after meeting them. It will keep you in their mind and provide a way to network.
A writer's business cards provide a way to network with others, both socially as well as professionally.