Dig a ramp down through the crawl space. This can be done using a small shovel and a wheelbarrow. Gradually dig gown and forward as you clear out space.
Keep going forward until you hit a support post. At this point, begin to focus on actually digging down to the depth where you would like your basement floor to be.
Dig a foundation for a new support post. Next to where the crawl spaces support post is, dig a hole that is about 4 inches lower than your basement floor will be and several inches in diameter. Fill this hole with concrete.
Place a new support post in this new area. The best one for this purpose is an adjustable steel support post, which you can put into place and then gradually extend until it is flush with the ceiling of the area you are digging out.
Continue digging through the center, repeating Step 4 as you encounter new support posts.
Dig out the periphery of your new basement. Don't get too close to the foundation (stay at least 2 feet away), or else dirt will begin to crumble in from around the foundation and you will need to build additional support structures on your wall.