Handkerchiefs are just as useful as bandanas, so long as you don’t use them for their intended purpose of blowing one’s nose. Wet wipes or some extra tissue paper can be brought along if you or the kids are feeling a little sneezy. The wet wipes and tissues certainly won’t be quite as messy as a handkerchief and can be put in a re-sealable plastic bag as garbage.
Read more about day hiking with children in my guidebook Hikes with Tykes.