I've seen this a couple times on Pinterest. "The words 'do not be afraid' are written in the Bible 365 times. This is a daily reminder from God to live every day being fearless."
Every time I see it, I think, "I always thought it was so people would stop peeing their pants at having a sword-wielding frightening angel show up on their doorstep. But sure, we can go with yours too." – Jo's Facebook Status
My friend unknowingly sent me off on a quest.
My short answer to "are there 365 'do not be afraid's' in the Bible" is no.
I have long since given up on trying to prove how many times the Bible does or does not say those words, however. Using the NIV concordance I began going through the Bible and learning from the times God, his Prophets, or his angels told someone "Do not be afraid." I read the verse in context, I searched for understanding how what it could possibly mean for us today, and I have learned so much that has guided me in so many ways.
My long answer now is it doesn't matter. The fact is God does not want us living in fear. He wants us to live in his love and grace, abundantly, standing firm in what we know is true.
I pray that whatever reason has brought you to search "how many times does the Bible say do not be afraid" that you will know He does say it. And not only that, the Bible addresses a multitude of fears through the different times He says it.
This site is no longer active, but I keep it here as an archive for those of us who may need reminding occasionally.