To use Lightning process builder do we need to switch our org to lightning experience?
No Process Builder works fine with Salesforce classic. No need to enable lightning experience for that.
The Lightning Process Builder is a tool that allows you to easily automate business processes using a convenient graphical representation of your process as you build it (It is more advanced version of workflow rules.). Automated processes in the Process Builder consist of:
Check out this short video to become more familiar with how the Process Builder works.
So there is no need to enable lighting experience it works without lightning experience as well.
Here is screenshot of Lightning process builder
To more about the process builder go through the lighting process builder trailhead module