There's really no reason to worry, but if you want to you might wrap some critical areas like the headset with something like this
stretch wrap that you can buy at Staples or most storage facilities.
As far as the headset goes, the thing to consider is where the gap between the moving parts is. Classic designs have the cups above, with a skirt extending below the cone so it naturally sheds water. However, in recent years designers started believing their own
PR about sealed bearings, and some headsets now lack the skirt that served for years, and open to the side
like this one, where the forward motion of the bicycle can drive water in. If your headset is of this design, you might want to use some wrap, or even a turn of electrical tape if driving through rain.