Yes a Swiffer Wet Jet is safe and great for cleaning hardwood floors as it doesn't get the floor too wet but does a good job at cleaning the dirt and lifting the dirt from the hardwood floors.
Another way to To clean hardwood floors is you can use some Pine Sol added with some mop water and then wring out the mop good enough to avoid getting too much water on the hardwood floors.
You can also use vinegar in a bucket of mop water to clean your hardwood floors and I sometimes use the Pine Sol and other times I use the Vinegar to clean the hardwood floors.
Just don't allow the hardwood floors to stay too wet otherwise you risk cracking or warping the hardwood floors.
So wring out the mop good before mopping the floor with it.
Vinegar is a very natural way of cleaning hardwood floors and other surfaces but Pine Sol is great for cleaning hardwood floors too.