I have an old walnut tree that I want to attach a knotted rope to for my boys to climb. Would the rope in the pic be a good choice? I am looking for something that would not hurt their hands too much and would be able to withstand some inclement weather.
IMHO that's much too small in diameter for comfort. Possibly also for safety in that material. The "safe working load" listed is almost certainly not one involving using the rope for supporting people or things above people, where it's typical to derate (or over-size) by a factor of 7 or more.
IMPE a 1 inch/25mm to 1.5 inch/37mm diameter rope is more typical for a "knotted climbing rope" in manila. That may be sold as 3 - 4.5 inch (circumference) rope depending how it's listed for sale.
I'm dubious of sisal for the job as it's somewhat more prickly (the major use I'm familiar with it from is as haybale twine - and that's hard on the hands.)
Plastic or plastic blend ropes will hold up to weather better than natural fibers, but can be damaged by sun over time.