If it is a stainless steel strap : To prevent rust, wipe off moisture, sweat or dirt with a soft cloth when convenient.  To wash the band, apply some water to metal bands / bracelets when stains on the surface are noticeable.
For heavy stains, use a tooth brush and remove the stain that has accumulated in the gaps of the band / bracelet segments . Protect the watch head from moisture entering the watch by wrapping it in plastic wrap etc. After washing, take special care to wipe the gaps of the band/bracelet segments with a towel and dry the entire watch thoroughly.
If it is a leather strap : Wipe off moisture and sweat as soon as possible by gently rubbing it with a soft dry cloth, and place it in a well-ventilated area after removing it from the wrist. Do not expose your watch to sunlight for extended periods (eg. on a car dashboard) since this may lead to discoloration/transformation/breakage of your watch’s leather strap.
If it is a silicone strap:  Silicone straps are more durable to the effects of moisture/sweat. Using a damp soft cloth clean the straps thoroughly and wipe it clean.