Grasshoppers are more attracted to your garden than they are to the basement, but some may take refuge there for various reasons. There's a good chance, though, that the little hoppers you see in your home's nether regions are crickets, especially if they keep you awake at night with their chirping. Either way, the best way to get rid of them is to take away what attracts them, which is food and moisture. You may also have luck with sticky traps and diatomaceous earth.
Grasshoppers are voracious eaters and are well-known garden pests, so if you have a basement infestation, it's probably because you're keeping plants on which they feed. Relocating those plants, if possible, will force the grasshoppers to move.
If the critters are actually crickets, they are there because they like the moisture in the dark corners and under the old tool cabinets. Because basements are cool and below grade, it isn't always easy to dry things out, but you can try running a heater or dehumidifier. In addition, conduct a deep clean that eliminates as many hiding places as possible, and seal cracks in the walls and foundation through which the insects enter.
Give Hoppers a Place to Go
When grasshoppers are a problem in the garden, a common tactic is to maintain an area of tall grass to attract them. Once there, they are often eaten by predators. This area may also lure grasshoppers out of your basement. Grasshoppers, after all, prefer to be in grass.
Repel Grasshoppers with Garlic
If the grasshoppers got into your basement, they probably know the way out, and you can encourage them to use it with a homemade garlic spray.
Garlic Spray Instructions
Mix 3 ounces of minced garlic with 1 ounce of mineral oil. Let the mixture steep for 24 hours, then strain it.
Add 1 teaspoon of fish emulsion and 1 tablespoon of Castile soap to 16 ounces of water.
Mix the garlic oil with the fish emulsion solution and put the mixture into a glass container with a closable lid.
Fill a plant spray bottle with 1 pint of water and add a 2 tablespoons of the garlic oil/fish emulsion mixture. Spray this mixture in all the corners of the corners of the basement in which you suspect grasshoppers to be present. It should force them back out into the garden where they belong.
Trap and Kill
When subtlety fails, you often have to resort to lethal measures. Perhaps the most straightforward is to vacuum the basement floor thoroughly -- especially around the perimeter of the each room -- and suck up as many of the little beasts as possible. Two other methods work when you don't have time for this direct approach:
Spread Diatomaceous Earth
Diatomaceous Earth is well known for its ability to dry out the carapaces of such insects as ants and roaches, and it will do the same to grasshoppers and crickets if you can trick the insects into walking through it. To this end, put on a respirator and spread the dust in all the dry corners of the basement the varmints are likely to traverse. Once the hoppers get the dust on their bodies, their fate is sealed -- the fine glass-like edges scrape the wax coating from their exoskeletons, and they essentially die of thirst.
Deploy Sticky Traps
If your basement is too wet for DE to be effective, you can catch grasshoppers and crickets with sticky traps. To make a simple one, wrap duct tape sticky-side-out around small pieces of wood or plastic and spread these around the basement floor. Check back frequently to refresh the tape and remove the trapped insects.