I have a 2002 Ducati Monster 750S i.e. and my air filter is directly under my gas tank, but you can see the air filter from just looking from the front of the bike under the tank. Is the air filter supposed to have a cover over it?
"supposed" is really a loose term. They come from the factory that way, Yes. However most people do some sort of mod to them either remove cover completely, carve larger holes, or buy the aftermarket performance filter. Either case, the filter gets change to a K&N. I removed the cover and used a square ring to hold the filter on. That was on my 00' 750. I have yet to do any mods to my S2R. If you start to allow more air in, then you need to make sure more air can exit. Otherwise she will run kinda crappy.
if your bike is running fine, don't worry about it.
are there any other mods done to your bike?