Dealing with water once its entered your basement is basically admitting defeat. Do you own this house or rent? What are the basement walls? Poured concrete or block? If its poured concrete and theres a crack, epoxy injection works very well from the inside (if you're diy savvy, you can find a kit for a little over $100 at a concrete supply company). If its concrete block or there is no evident clearly visible crack in poured concrete, then you'll need to address seepage from the outside- which means digging, then applying a membrane to the outside of the wall. And, while you've got a hole dug, you might as well do an exterior french drain. You can try something like the "Drylok" paint products on the interior, but they wont stop much.
Properly routing downspouts is the most common and cheapest fix. Put on your rain coat and go out and observe during hard rains. If downspouts are piped underground, make sure the pipe is actually taking the water (i.e. it not spitting up and out of the pipe). Find where your drains discharge-usually either through the curb into the street or into a nearby catch basin. Make sure that the water is actually making it to the discharge. If its not, that could indicate a clog or break somewhere underground. If you think you have a clog or break underground, get some pipe that you can lay on top of the ground, pointed away from the foundation and use that as a temporary fix-check the basement and see if it helps. Your goal is to give the water a better place to go than into your house. If your underground piping is clay tile (probably really old) or the asphalt impregnated paper shit they used in the 70s (just total junk), then its a safe bet that the pipe is either clogged or even collapsed. If its pvc, rent a camera snake, and see if you see any troubles. You want to chase drains from the point of entry all the way to discharge. Sometimes, people halfassed stuff and just ran a pipe a couple 10, 20 feet and just let it dump into the dirt- that can be fine if theres a drywell or you have really loose well draining soil, but around here (lots of stiff clay) its pretty much useless.
Also, make sure grade is sloping away from the house- they say for at least 10' around the perimeter. Make sure that water has a swale or path to follow away from the house- preferably out to the street or to a creek (or your neighbors yard, lol).
Don't accept defeat.
