I didn't for a while, but the issue hit home about a year ago or so when a local realtor was struck by a car while riding his bike over in a nearby town in Vermont. He was brought to a hospital but without any ID was simply there as an unknown patient until his wife reported him as missing and was eventually connected to the hospital. I don't recall exactly all the details, but I do know that it was enough to make me change my ways.
Now both my wife and I run, walk or bike wearing a bracelet like the one I show in the photo (that is, in fact, mine). It is simply a plastic case that when you open it up contains a piece of folded paper with my name, contact info and medical info on it. Enough that were something to happen to me, someone could reach my wife or other emergency contact and also have basic info that would help in getting me care.
The particular bracelets we bought were from ID On Me, but a quick Google search will show you many options. Before this, I used to carry a photocopy of my driver's license in a fanny pack... but as I started running I stopped wearing the fanny pack, so for a while I was ID-less.
If you do carry ID while exercising, what have you found that works for you? If you don't, why not? (And could I suggest considering it?)