Purchase or rent a humane trap. The HavaHart trap manufactured by the Woodstream Corporation is the most commonly used by professionals.
Read the directions carefully to understand how to operate the trap.
Bait the trap with peanut butter or a mixture of oatmeal and peanut butter. Most rodents are attracted to the smell and taste. Because of the texture of peanut butter, it is not as easy for an animal to grab it and run.
Place the trap in a location that is easy for the squirrel to gain access to or where you have seen evidence of squirrel activity or droppings in the basement.
Leave the basement. Check the trap every few hours to see if the squirrel has been trapped.
Check with animal control authorities in your municipality for information on what to do with the trapped animal. Some communities impose a fine for releasing a squirrel locally, so don't be tempted to let it out in a nearby park. Squirrels are also territorial. If released nearby, they are likely to return.
Put on protective gloves. Check the basement thoroughly to be sure that the squirrel has not left offspring behind.
Look for the area where the squirrel entered and make repairs. At a minimum, nail a length of hardware cloth over the entry hole until you can make more extensive repairs.