I was recently asked to not come to office for a few weeks due to the coronavirus situation. My state Ohio is offering unemployment insurance and benefits to all employees, however I am not sure whether I should (or can) avail such benefits and would it impact if I apply for H1B or Green card in future as I read a recent rule about green card applications may be denied if using public benefits.
You do not indicate what your current status is. More information is needed to fully address your question and your situation. Accepting unemployment benefits in and of itself is not an issue for immigration purposes.
Generally, unemployment insurance benefits are not available to nonimmigrant visa holders. Check with your state UI Department.
Worry not and accept in full confidence - state unemployment insurance benefits is NOT "welfare" by any stretch of the imagination and will not have any negative effects on your current H-1B status nor "green card" application in the future - will not need to even disclose it.
Unemployment benefits is not welfare. Nothing to worry about.
