The short answer: A mitigation sealer can resolve the moisture issue.
The problem: You should find out why you have the issue first. If you have a poor or no vapor retarder, you are going to get moisture movement up through your concrete slab. This moisture can be stopped using a high performance mitigation system, or slowed by using a less effective moisture blocking system.
Some systems protect to a certain level of moisture emission, say 8lbs of MVER or 80% RH. When you put these systems down, you are hoping the moisture level in the concrete slab never exceeds the threshold of the moisture system. If it does, the system may fail and you will be right back where you started except you will be that much poorer and wiser.
Other systems protect to an unlimited moisture and alkalinity level. These obviously will be more costly.
If you find your slab has a working and effective vapor retarder and there is no chance that the slab will get any additional moisture gain, it may be feasible to use a less expensive mitigation product that provides some level of moisture blocking. If however your slab has had 20+ years to dry out and still causes flooring to fail and mold to form, chances are good that the moisture problem is much larger than just excess mix water needing time to dissipate.
In your case I would recommend against a limited system such as MVP and suggest a moisture mitigation system designed and warranted to unlimited alkalinity and vapor transmission.
You will be told by many that this or that product will work just fine, and you will be shown warrantees and guarantees and dozens of happy clients, but you must think about what the cost of failure is, considering mold and having to leave your house, let alone the failure and replacement. The truth of the matter is, if moisture is getting into the slab and you put something on top of the slab that seals it (being wood, vinyl flooring, or even a mitigation system) the moisture level will start to go up over time. At some point the moisture and alkalinity will defeat 99% of the systems out there.
If I were you, I first look for the cause of the failure, and then either install a bulletproof mitigation system or go back to a breathable floor covering.