You mention that the wizards can't use their magic if they aren't wearing their ring. You also say the soldiers don't know this particular fact, but do have reason to suspect something supernatural.
During daytime hours, when the museum is open to the public, have all visitors remove their jewelry. While it does inconvenience several people, the risk is worth it. (Of course, they'll need a way to secure the jewelry, but that isn't part of the answer to this question.) The soldiers can then detain anyone wearing jewelry for interrogation.
Some reasons for the public why jewelry is banned:
1. Because the sign says so. How often do you question why certain signs are posted in public facilities? When people see a sign, they tend to follow it, regardless of how logical it is.
2. There was an assassination attempt involving poisoned jewelry and the intended victim is in the museum again and asked for no jewelry. A little more far-fetched, since such an event would probably find its way to the news, but it also gives a reason why there are soldiers in the museum.
3. There have been reported events of people wearing jewelry (and only those wearing jewelry) falling unconscious since the artifact came to the museum. 619
Out of an abundance of caution while the effect is investigated, jewelry is being banned for the protection of the public.
When the museum isn't open, use the typical sensors museums are equipped with. Have soldiers perform random patrols, changing their routes at the flip of a coin (human decisions aren't random enough). Have guard dogs inside and outside the museum. Their senses should be sufficient to detect approaching intruders. The artifact itself can be put inside a box anchored to the floor and secured with a combination lock.
In The Da Vinci Code, we are introduced to an item called a cryptex. This is a device secured with a combination lock. It contains a piece of papyrus and a vial of vinegar. If someone tries to open the cryptex by force, the vial breaks and the vinegar destroys the papyrus. For another layer of security, you could do the same thing with the artifact and its box, but with a powerful acid (supposing acid would work). 619