Should our Governors have the right to "detain" people with a contagious disease or had CONTACT with a contagious person?

There was a time when immigration into this country was not permitted if you had any illness or communicable disease.  That has fallen by the wayside.  Do you really want people coming into this country or your state that have communicable diseases?  How about Leprosy, is that OK?  How about tuberculosis?  What about the new strain of Covid 19 that is more highly contagious than the old one?

In my view, if they have the disease, yes.
If it's just contact, it should be testing and self-isolation for 14 days.

In Australia, we've been using strict quarantine here since the start of the epidemic.
The first problem was inadequate guards - but once we learned how some people will try to escape, we learned to use police and stricter supervision.
Our system uses specified 5-star hotels for quarantine at the person's expense, but if they progress to serious symptoms they're instantly moved to hospital where the treatment is free.
It has prevented our hospitals from being overrun.

But on it's own it's not enough.
Widespread free testing and self-isolation for 14 days after a test is also necessary, as is total lockdown of hotspot areas.
The rest of the country uses social distancing, good hand hygiene, and masks when closer contact is unavoidable. In areas where there've been no outbreaks, most people don't bother with the masks.

All this quickly contains outbreaks, brings the numbers down, and keeps the rest of the country in work and functioning in a relatively free and normal way.

If you feel like it, check out the history and statistics on our databases and see for yourself. 

  Who has the authority to enforce isolation and quarantine because of a communicable disease?

The Federal government derives its authority for isolation and quarantine from the Commerce Clause of the U.S. Constitution. Under section 361 of the Public Health Service Act (42 U.S. Code § 264), the U.S. Secretary of Health and Human Services is authorized to take measures to prevent the entry and spread of communicable diseases from foreign countries into the United States and between states.

The authority for carrying out these functions on a daily basis has been delegated to the Centers for Disease Control and Prevention (CDC).

Under 42 Code of Federal Regulations parts 70 and 71, CDC is authorized to detain, medically examine, and release persons arriving into the and traveling between states who are suspected of carrying these communicable diseases. As part of its federal authority, CDC routinely monitors persons arriving at land border crossings and passengers and crew arriving at U.S. ports of entry for signs or symptoms of communicable diseases.

When alerted about an ill passenger or crew member by the pilot of a plane or captain of a ship, CDC may detain passengers and crew as necessary to investigate whether the cause of the illness on board is a communicable disease.