There should NEVER be water in the basement, unless a pipe breaks or the sewer backs up. Rain -- even a LOT of rain -- is NOT a good reason.
Putting towels down and leaving them will make it MORE likely to mold. if there is water to mop up, mop it up and remove the towels.
1. If you have no plumbing issues, the water be groundwater, coming through the foundation from the bottom or from the side wall.
2. You could have downspout issues. Your house should have gutters, and all the gutters should have downspouts. Check to make sure you don't have a downspout sending rain water straight down the side of your house. Every downspout should have an angled spout that carries the water out at least six feet away from the house.
3. You could have yard drainage issues. Make sure there is no place in your yard where the ground slopes toward the foundation; Walk around the house; the earth should be graded so that it slopes AWAY from the house, all the way around If this is not so, you may need to install an underground drain system to carry the water out to the street.
4. If you can't see an obvious problem... You need to have it evaluated by a professional.
By the way, unless you have a flood policy.. water coming in like this from groundwater, or from a downspout, or from poorly graded property, is NOT covered by homeowners insurance (because it is considered flooding).
THEREFORE, the sooner you get the problem resolved, the less it will cost you.
If you re a renter- notify the landlord RIGHT AWAY.