Does the white sport band get dirty easy?

I can only speak from my own experience. I've had the watch on a white sports band since launch day and it looks clean and bright to me.

I wear it every day - work, running, gym, home and don't take any particular precautions.

Other people have reported marks and stains, but it seems they come from spills and skin treatments (oils and creams)

It will get as dirty as anything else exposed to dirt, but white obviously shows it more easily than anything else.

Fortunately it is easily cleaned. Read Cleaning and wearing Apple Watch - Apple Support and scroll down to "For all other bands, buckles, and closures".