This is a long-standing debate between writers, the answers is yes, and no. There are jobs on job boards that do require a degree and others that require an advanced degree. What you need to write for clients is a good working knowledge of English spelling, grammar, and punctuation. If you are not up to par, take a course at a community college or university. Find a grammar book, read it, and do the exercises in it. Read everything you can get your hands on. You need to have the basics of the English language down before you can write for a client. Clients expect perfect English in their assignments for freelance writers.
The elitist writers think you should have a degree in English or Journalism to be a freelance writer. If you have one, fine, but it really isn’t necessary to pick up writing assignments. You do need to be a good writer, which means using the language correctly and always striving to improve your writing with every piece of copy you write. This also goes for all writers. If you are ever satisfied with your writing, you will become stagnant and become lazy in your writing. Your writing will reflect this, and the client will notice.
A writer needs to be creative in their writing, try to eliminate passive sentences, watch verb subject agreement, typos, grammar, punctuation, and possibly keyword density if the client wants SEO web content. You can learn all of this. It just takes time and effort on the part of the want to be freelance writer. With practice and reading, studying how writers put sentences together, and learning or refreshing the basics of the English language, anyone can become a writer. This freelance writer knows a person that is dyslexic and she is a published author. It does come down to how bad to you want it.
Yes, a degree is a nice thing to have, if you can afford it. If you can’t, read grammar books, Basic English books, try taking a course. The degree does hold more weight when you apply for a job. Your writing skill will hold all the weight you need for most jobs. What a writer needs is a portfolio of well-written copy they can show to a prospective client.
If you want to become a freelance writer, learn about what freelance writing is about, and make sure that you are up to date on the rules of English; for which there are many style guides. If you plan to submit articles to magazines, make sure to read the writers guidelines. You would be surprised how many queries and stories never make it because of writers submitting the wrong material or not following the guidelines completely.