Flush with white vinegar to get rid of bacterias mildew etc.
Reflush with water then refill with strong concentration of washer fluid.
Use washwipe a few times per week if possible.
Never had a problem with living organisms in the washer bottle with any car, until my current E46 and E86!
I have cleaned out the reservoir and filters on both in the last year - all I would add to the previous advice is to keep topping up with the SAME concentrate (as opposed to concentrate in general) because in these cars different brands of fluid seem to make the problem worse.
Yup as said above, it is indeed crap growing in the system. Mine started doing this a year or so back. whole system has been flushed out and now I make sure that it's used every time I take the car out, even if it's a bone dry sunny day. Plus keep it topped up with well concentrated fluid. Been fine since. Just need to clean the Blue stains off the bonnet when you use it on a dry day.
My daughters 1 series smelled like a sewer when we purchased it. Ran some water through it and put a Milton sterilising tablet in it. Drained it and now keep it filled with correct concentration of screenwash. Never a problem since after a few years.
No need to keep using the jets each day or week as it just messes the car up.
Don't mix screenwash types though as mentioned above. They tend to create a jelly that clogs the filter
Run it through all the pipes inc. headlight washers if you have them. I'd then drain it fully (just pull the pump out) and run your diluted screen wash through all the pipes, so you are sure you are all set and primed with screenwash.
Talk about bacteria or whatever in windscreen wash bottle I have just been looking in our central heating header tank in the loft and there is a hard crust with terrible looking growth around it. Donned a mask and latex gloves, drained it down and scrubbed the tank with bleach. Just proves that bacteria will grow anywhere.