Water Damage
If you get the top of your DSi wet, liquid can seep in through the small speakers or in between the buttons and destroy the internal components. The potential damage varies depending on the amount of liquid spilled. A small amount of liquid on the screen, for example, will not result in immediate damage. Alternately, if you spill a significant amount of liquid on top of your DSi, it could quickly spill into the system and cause partial or complete damage. Depending on where the liquid affects the console, many problems can arise. You may notice some of the buttons have stopped working. Your battery can get damaged, the built-in camera may stop working or the system will not turn on at all.
Damage Control
If your DSi is exposed to a small amount of liquid on the top, wipe it off with a soft cloth. Do not use any abrasive material that might scratch the screen. If a significant amount of liquid is spilled on top, turn off your DSi immediately and remove the battery to prevent damage. To do this, fold your DSi shut and flip it over so that you can access the back. Use a small phillips-head screwdriver to remove the two screws on the battery cover and remove the battery. A useful technique is to open your DSi and submerge it in a bowl of dry rice. The rice will help remove moisture faster than if you leave the system out to dry. Do the same with the battery.
Warranty
Unfortunately, Nintendo's warranty does not cover water damage. If the top of your DSi gets wet and damages your system, you must either send it to Nintendo for repair or purchase a new DSi. Sending your DSi in for repair can be a hassle, and they may not be able to fix the problem.
Prevention
The best thing you can do is prevent water damage from ever occurring on your DSi. Keep the system in a dry place where liquid cannot reach it. Avoid drinking anything while you play with your DSi. If you drink something while playing, keep a soft cloth handy to deal with any unexpected spills.