Brake dust isn't only unsightly. It's corrosive to your wheels and inevitable because brakes require friction and the wearing down of your brake pads to work, so your wheels will always be exposed to dust from your brakes. It's designed that way. The dust on automotive wheels is 92% iron at that, so the dust tends to wear down not only your wheels and its rims and hubs but also the cast iron brake rotor as well as the brake pad's semi-metallic elements.
Therefore, as a responsible motorist, you should devote some time and effort in making sure the dust is cleaned off, especially in light of the remaining 8% of the dust being composed of carbon content that's also wheel-corrosive.
Remove Build Up to Lengthen Wheel Lifespan
Buildup and Corrosion: The dust from your brakes is corrosive and builds up on the wheels. Leaving them on your magnesium or aluminum wheel is like leaving iron wet. You're asking for trouble and damage that way. Instead of rusting, the corrosive elements etch unto the finish of the wheel like so much abrasive material from sandpaper, causing discoloration, peeling of paint, and staining if you're not careful.
Fine Auto Finishing Protection: In order to protect the finish of your wheels, hubs, and rims as well as the finish around it (as in the paint finish of the car itself as the brake dust starts to spread around if it remains unattended), you need to wash and dust them off like the usual house dust. This will prevent future degradation from discolored wheels or even outright pitting that lowers the value of your car when you decide to sell it or trade it in. Wash then wax is your best bet.
Professional Detailers versus Local Car Washes: Local car washes are more concerned about cleaning your car as fast as possible and making it look good in a showroom kind of way at a glance. However, if you look at minute details (like the state of the wheels and hubcaps), that's where you can see the difference between a quick trip to the car wash to professional detailers that gets your car looking perfect by getting rid of dings and cracks on the surface on top of brake dust corrosion on various types of wheels and their finishes.
Detailing Will Cost You More: Detailing isn't cheap. DIY cleaning while paying attention to the brake dust buildup is much cheaper. The latter is certainly preferable than the former, especially when you just got your car and you only need to maintain its good condition while its mileage is still low. However, you will need detailing in order to keep your wheels in good condition after a lot of mileage has come to pass and you've neglected to properly clean the wheels.