Minecraft wiki entry for Survival_in_an_infinite_desert, associated text: Mark a central point with a tall marker, facing west, so the setting sun casts a shadow hopefully.
Mark off a distance from the central marker, on either side equal to the depth you need to go down. Don't get this wrong. Add a little extra (not much), if you are unsure.
Each day dig down an equal amount on each side.
Keep a careful watch on the sun and make sure you can get back up in good time.
Build one or two tiny emergency survival shelters leading off the side of the staircase in case you mess up (at least 4 blocks away from the staircase, but can be just 2 high by 1 wide).
Dig only a one-block wide staircase (to prevent spiders spawning or entering your staircase).
Wall-off both of the top entrances to the staircase, and leave sand pillars behind you along both 'legs' of the staircase at intervals to stop anything sneaking up behind you (but be prepared to remove them quickly if a mob comes from in front of you. Leave these walls and pillars there from day to day. Just destroy and then replace them as you move through.
Once you've cleared the sand on both 'legs', you need to mine 312 blocks of sandstone (52×3×2). By hand, at 4 sec per block requires a minimum of 1248 sec, around 25 minutes. Expect to use up several full days on the project.
Once you are finished, remove all the pillars and open up the walls. Then, entice a creeper to chase you down the staircase. At the bottom, turn around and provoke the creeper into blowing up right at the bottom, on the exposed stone. Timing is everything.
Repeat indefinitely, until you have mined all the cobblestone you need, or all the cobblestone in the world, or (more likely) died due to a timing error.