I had a glass container with turmeric roots. When I was preparing curry I noticed that they have sprouted but seems to be firm and not problematic in other way (no mold etc.). Is it safe to use them?
Yep, it's perfectly safe. The only time you should discard it is if mold has formed, or if the roots give easily when you squeeze them.
You can store them in the fridge in an airtight container along with a desiccant pack to prolong shelf life, but much like any other root they're at peak flavor within a few days of purchase.