What will happen to 1GP packages if we turn on "Enable Unlocked Packages and Second-Generation Managed Packages"?
Nothing. But there's an assumption here that cannot be correct:
my company has a dev hub, that owns all our 1GP enterprise managed/unmanaged packages.
Dev Hubs do not own 1GP managed packages - the packaging org does. Your Dev Hub may have the namespace linked to it, and that is fine.
Can we guarantee that nothing will break and there is nothing to worry about?
You can never guarantee anything, but turning on 2GP is a pretty safe change. It shouldn't impact anything else in your org unless you have some integration that is exceptionally sensitive to seeing new objects in the org's Tooling API schema.