How can I repair a basement floor that sometimes gets wet?

You can paint. Go to your local stand alone paint store. They can advise you best and hook you up with the right products.

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Paint won’t stick if the floor is wet. Is the moisture coming up through the concrete? Or are the walls damp too?

Stopping the water infiltration is key on this project and for the health of your entire home and you.

I’m assuming your basement doesn’t have a sump basket and pump and if that’s the case the first thing I would do is check outside along your foundation. Make sure the grade slopes away from your home so water is directed away from the foundation. You may need to add some drain tile and pea gravel to help with this depending on your situation, and if you don’t already have them gutters will help too. I would head outside during a good rain and watch what the water is doing as it’s falling to help you find any problem areas. Personally I wouldn't try to do anything to your floor until you’ve stopped seeing any water for several wet periods and it has had time to thoroughly dry. Best of luck to you!

We do have a sump pump and that's working fine. We moved the hose to the other side instead of the front which seems to be helping. It is drying out but it is supposed to rain again on Friday. We will have to watch with happens then. We also have gutters. I agree with you about not doing anything now until it dries and doesn't start again. Thanks for your advice.

Since the painted floor is peeling, you might consider concrete stain/dye.

It will not peel as it literally dyes the concrete.

It can be rolled on or sprayed on to create different effects. I use a faux sponge roller cover.

It comes in transparent & opaque formulas, depending on whose brand it is.

It can be thinned with water, if you wish.

Hydrostatic pressure is hard to fix once it starts - I’ve experienced this in two other houses’ garages.

All the rain drainage suggestions are great.

Good luck.