How to seal moisture due to reported unsealed put covered cement slab

Sounds like you may need dehumidifiers.

What are you planning on putting down on top of the concrete for flooring? If you are puttin something on top, I would just go with sealing with Dryloc... https://www.youtube.com/watch?v=0yACqGYROG0

If you are planning on using a stain, I would seal with whatever chosen products' own pre-treatment sealer.

If you leaving as is Behr has a clear waterproofer... https://www.youtube.com/watch?v=blcMqx5LX8c

I would address the mildew in the closet with a mold cleaner or vinegar prior to sealing and make sure its dry, put a fan on it.

Hello Anne, I am SO sorry to hear about the mildew smell in the closets due to the unsealed (but covered) cement slab on the lower level. I used to live in a basement apartment and sometimes my carpeting would be wet for "no apparent" reason! The smell sometimes would be horrendously dank and musty, and all sorts of little critter bugs like I've never seen before would make their way into my apartment. I ended up moving out after a year, when my lease expired. I know that DEhumidifers are HUGE energy users, but it may be your only solution to bringing down the humidity (and odor) in the closets, and in the basement. If you own the home this article may be useful to you, and if you don't own the home you should bring up the problem to your landlord and show him/her this article:

https://www.csmonitor.com/1981/1120/112030.html

http://www.greenbuildingadvisor.com/blogs/dept/musings/all-about-basements

last but not least (and maybe Ninzerbean's solution could be yours too... perhaps check out Spycor Building Products after reading the article) ...

https://www.instructables.com/community/How-can-I-get-rid-of-the-smell-of-damp-concrete-in/

Good Luck, Anne, and thanks for asking HOMETALK about how to deal with the moisture (and odor) caused by the unsealed but covered cement floor.

Thanks so much and I will follow up on your resources!!