I'd argue that they already do in some basic ways: no jurisdiction allows 14 year old to vote, for instance. We're _supposed_ to enforce citizen-only rules, too. Adding a uniform set of basic integrity requirements would level the playing field nationwide... that's the intent.
The Constitution limits its "interference" into state elections only regarding qualifications for office (e.g., age of Congress members, age/residency/natural born for President, age of "electors").
Beyond that, elections are state affairs. See my dedicated post on your overall topic.
-PJ
Who is going to pay for all of this?