What - specifically - do you think the two have to do with each other?
No, seriously. The kind of cybersecurity issue you're talking about applies at the level of the individual systems hooked to the internet, with security provided by a firewall or other connection. Not by "control" of the internet itself.
If the issue is that internet-connected critical systems, whose failure can harm Americans, currently have inadequate protection against attacks, then why would you imagine it's only "Demcrats" who would demand something be done? Personally, I'd be opposed to any politician of any party who stood in the way of it.