Usually the dirt is exterior...the interior is usually fine. If the cover is the silver coated stuff, it is like ironing board stuff...and funny thing is, once a year I would use Palmolive green dishwashing soap (just a tablespoon in a bucket) and clean it while on the bike (yeah yeah, just be quick about it and rinse gently with hose after), I then take it off and hang it on a clothes line, usually on a clear day not in the high noon sun and heat, just to air dry.
Then I go to cleaning the bike...which may get a little wet, but did not get direct hard water splash, so I can clean the bike up that much easier. BTW, the covers will only last for about 6-8 years anyway (the fabric usually wears thin in places) from sunlight and UV...so you might want to try some spray preparations on it like ScotchGard to help it last a few years longer...