Good question. The answer here depends on several factors, including where you work and what type of writing you do.
Let's start with the simplest factor: where do you do your writing? If you write from home, don't own a lot of "business property," and clients do not come to your home to do business, then you won't need business insurance in the traditional sense (but you do need health coverage, so check out Working a Freelance Job? Consider These 3 Health Insurance Options).
But let's say you rent an office space to do your writing and you have clients come visit from time to time. If that's the way you do business, you should look into getting business insurance to cover your business property (such as furnishings) and on-premises liability (in case a client slips and falls on your premises, for instance).
Now, with the simple stuff aside, let's get into the meat and potatoes of the issue. The biggest reason professional writers should consider business insurance is for the liability risks that might come with the job. Just how high that risk is will depend on the industry you're writing for. If you specialize in writing about home decor trends, there's a good chance you'll never find yourself at the wrong end of a lawsuit (at least, not over the content of your writing). But if you're a medical writer and people rely on your content to make medically informed decisions, then your liability risks are pretty high (see Liability Risks on Social Media for more liability risk scenarios for writers).
For journalists and professional writers, the potential for defamation is the main concern. When we talk about defamation in the insurance context, we mean any statement that causes unjustified injury to the reputation of another person or business. The key here is that it must be unjustified. If your damaging statement can be proven to be true, then you have not committed any defamation. For example, exposing a corporation's illegal activity doesn't really qualify as defamation, even though it will likely damage their reputation.
Depending on the type of writing you do, you might be more susceptible to being sued for libel (written defamation) than others (regardless of whether the suit is justified or not). We've all seen the headlines – journalists and reviewers are notorious for being the subject of libel suits. If you are a journalist, a reviewer, or someone who frequently tackles controversial or high-risk topics, you should almost always purchase business liability insurance.
Even if you take the utmost care to publish only true statements and are right 100% of the time, the cost of defending yourself against a lawsuit can be devastating to your finances and your reputation. With professional liability insurance, not only does the insurance company pay for any damages, they will also pay for your defense.
While your standard Commercial General Liability policy would likely exclude journalists and writers, there are professional liability policies available for writers. And with premiums these days as low as $300/year, there's no reason not to be protected.