We left our grass. I love it, but "trimming" around the playset is a pain and we do have minor wear spots under the swings and at the base of the slide. Regardless, I like the look of the grass.
The logic behind the mulch is that it will cushion the falls. I don't know how much the rubber is, but I think it would be great. Just make sure you frame out the area, put down a landscape fabric to prevent weeds from coming through and then put down the mulch base.
I don't think wood would poke an eye out. You can get different colors and quality of mulch (different sizes). There is a cheap one - that is really just tree branches ground up - very stick like. Then there is true mulch which is more wood chips. And then there's a shredded variety.
I have no doubt the wood would be cheapest, but needs to be added to every couple of years. Just make sure you frame it out and create the right base. It's more expensive up front, but well worth it in the long run.
Also, consider "sealing" the playset if it's wood. Ours was the Rainbow brand. They say the stain is weatherproof - WRONG. It showed so much wear after just one year.