I have the same problem. I reported it on their github repository but they haven’t responded yet.
@expofield @tiimba
Please select Menu Style Layout as Sidebar or Fullscreen to avoid closing of the menu in Appearance -> Customize -> Header -> Menu -> Mobile Menu
Hello @prajakti, thanks for the reply.
The sidebar and fullscreen option are also not usable on mobile, because submenu entries move away when you try to tap on them.
See my bug report on github:
https://github.com/cyberchimps/responsive/issues/92
@tiimba
Thanks for your feedback. This is a known bug and will be fixed in the upcoming release.
@tiimba @expofield
The menu issue on mobile has been fixed in the latest release of the Responsive theme. Please update the Responsive theme to the latest release v4.2.1
Thank you.