If a person wants to be really, really good at something, so good that they want to make money from it, whether it be writing, gardening, cooking, sewing, or painting, should they forsake all other interests and expend all their energy on that one thing? This is in addition to the daily responsibilities of work and family. I tend to think sole concentration may result in obsession, but in the pursuit of sucess would obsession be a good thing? Or would a second thing offering a balanced, sane distraction have its merits?
My immediate sense is that balance is best in all things. With writing, I think it would be especially good because being involved in stuff other than writing gives you a fuller world to look at, fuller ideas, perspectives, and such. It gives you stuff to write about.
Having various interests, I have found they all fuel my writing. Music, photography, reading, the study of other countries. Everything comes back to my writing, I can pull something from those hobbies. If I set those things aside, I know I would abandon my inspiration.
Writing is rare in that you can, as Chimmy stated, incorporate all other interests into it. However, if you truly want to make it your profession, you must approach it with the same level of commitment you would any other job.
I must agree with Marina that balance is a crucial component. To have a variety of interest and ample time to explore them can greatly enrich your writing, but it is also quite important to set aside some substantial time to get writing done, especially if it is in a professional capacity.
Writing, more than any other avocation I can think of, thrives on a liberal education, which is to say one which covers a broad spectrum of knoweledge categories. I don't simply mean education in a school, either. Most successful people I know study even more when they are out of school than they did working for a degree. They hust don't have to write as many term papers. My interests range all over the place, from physical sciences and mathematics to psychology and sociology, from photography and painting to hiking and snorkelling, from speculative fiction to cooking. And a lot of humor and wordplay. All these interests and more impinge on my writing. Follow and develop every interest you can. You don't have to be an expert in all of them, but the more knowledge and life experience you have, the happier you will be. And that is true regardless of whether you choose writing as a concentration area.