I drive 55 miles one way to my son's home to babysit my 2 grandsons. I do this 4 days a week and I also do their laundry and housework. He pays me weekly for this. My question is, do I have to pay self-employment tax (SS & Medicare taxes) on this income?

Yes, if you want to treat this as income. How is your son treating this payment to you? Did he file 1099-misc?