If you do not want the hardwood floor to get wet or come in contact with any chemicals from the dusting agents, then you may consider vacuuming the floor. In fact, vacuuming is the best way to dust hardwood floors. Vacuuming can not only remove dust but also other things like hair wood powder etc. You can do it with any kind of vacuum cleaner. No need to spend a bulk amount of money and also you won't need any kind of special attachments.
Hardwood floors easily catch dust particles. Nobody likes a dusty floor. In this case you might think that sweeping is the best way to get rid of these dusts. But it will further spread the dust everywhere. So sweeping is a bad idea. You can use a mop to easily clean off the dust. Spilling some water will increase the effectiveness of the mops dust catching ability. You can add a dusting agent in the mop if you want. Just make sure it does not contain any harmful chemical for hardwood.