It's totally safe to buy a refurbished phone, but it depends on where you're buying it. There are both good and bad refurbished phone stores, so you need to choose the store carefully and ensure that they provide a warranty and keep their promises. There are so many who use Laser marking machine and some use their tradiotional methods to refurbish. And some store replace phone screen or other parts. 

Hi Stan. I think generally, it's safe if you buy one from those stores that give warranty. Personally though, I'm not a fan though. I'd rather buy a cheaper phone as long as it's brand new or hold on the purchase 'til I save enough.