If water is leaking into your basement low on the walls or at the seams where walls meet the floor, your problem is hydrostatic pressure pushing water up from the ground.
First, check whether you have footing drains, underground pipes installed when the house was built to carry water away from the foundation. (Look for a manhole or drain in the basement floor or a cleanout pipe capped a few inches above the floor.)
If the drains are clogged, open the cleanout and flush the pipes with a garden hose. If that doesn’t work, a plumber with an augur can do the job for about $600.