I own a small apartment block, and I clean mildew off of silicone caulk all the time.
If the mildew is on the grout, you're best bet is to use phosphoric acid to dissolve the surface of the grout until the grout is clean, and then use a grout sealer to seal the surface of the grout so that it doesn't come back.
If the mildew isn't too bad on the silicone itself, you can clean it up by mixing Borax (which you can find in the laundry detergent aisle of any supermarket) with bleach to make a thick slurry. Use a large spoon to scoop up that slurry and hold it near the silicone and then a small spoon to spread the slurry over the silicone. Then, cover that with Saran Wrap or any cling wrap for a few days. When you remove the cling wrap, the silicone caulk should be white as Manitoba snow.
Now, in the above instructions, you'll get the same cleaning effect regardless of whether you use Borax or any powder since the active ingredient is the bleach. However, my experience is that using Borax makes for a stickier slurry that will stick to vertical surfaces much better.
Also, if the silicone is really badly mildewed, the above procedure will only lighten the mildew, and you'll still have brown stains on the silicone. In that case, the only option is to replace the silicone.
It's common to have people who have replaced the silicone around their tub find that the new silicone caulk won't stick. The reason why they're having a problem is because they haven't removed the old silicone caulk COMPLETELY. So, what they're trying to do is get the new silicone to stick to a very thin film of old silicone caulk that is still on the tub and tile. THE BEST WAY to remove the old silicone is to:
- Cut off as much as possible with a single edge razor blade gripped in the jaws of a pair of needle nose style locking pliers.
- Apply "Silicone-Be-Gone" silicone caulk remover which you should be able to buy in the caulking aisle of any hardware store or home center. Silicone-Be-Gone is nothing more than gelled mineral spirits. It does not DISSOLVE the silicone caulk, but causes it to swell and become quite soft so that it can be more effectively removed by mechanical means.
- Scrape off as much of the swollen silicone with the razor scraper described in Step #1.
- Repeat Step #2 and then scrub the area with a green Scotchbrite pad of the kind sold in grocery stores as pot scrubbers. (If you have a plastic tub or shower pan, use the white Scotchbrite pads for delicate scrubbing. The white Scotchbrite pads don't have any abrasives impregnated into the nylon fibers from which the Scotchbrite pad is made.
Silicone-Be-Gone will emuslify in water, causing the water to turn whitish. If you've removed all of the old silicone from the tub and tile, the new silicone caulk will stick to both the tub and tile like chewing gum to the underside of a church pew.
And that's it. Clean as much of the mildew off the grout as you can with phosphoric acid, which you can buy as a toilet bowl cleaner from any home center of place that sells janitorial supplies. Then clean the silicone as best you can with a borax/bleach slurry or remove it entirely using Silicone-Be-Gone and replace it.