How can I clean my 3DS?

Nintendo says you should use a dry toothbrush. Not particularly helpful in the stickiness department, though.

What to do

There are a couple options for you. Firstly, you need something wet to clean it with. Best bet? Isopropyl alcohol. Windex and other solvents of that nature works, and Goo Gone is probably alright too. Warm water also works, if you have that.

Next, you need a cleaning implement. Best bet? Toothbrush. Q-tips work and so do cloths, but they have to be used in tandem with the toothbrush. Don't spray any of the cleaning solvents onto the 3DS. You need to only get the implement "slightly" wet. You would dip the toothbrush into the solvent and shake off excess, wet the top part of the q-tip, and mist the cloth with the cleaning solvent.

Do not use: Tissues (they leave residue occasionally), paper (sand paper or otherwise leaves scratches on the system).

You want to make sure your system is off and carefully clean the slider track. Leave it alone until the solvent dries, then come back and move the slider. Clean again and wait for the solvent to dry again.

Will this work?

Depends. It sort of depends on the "damage". If the tracks aren't the only thing that is sticky, you might have to replace the controls. That's a long process. It might be better to use a toothpick or a pin in that case to try and shake out some debris.