I carry band-aids with me all the time. They're usually in my backpack inside a small, zippered bag (in my case an old makeup bag) and I wouldn't leave home without them. The reason I bring them everywhere is that I need them a lot. My kids' knees, hands, elbows, fingers and feet just seem accident prone, whether we're on a hike in the woods, at the playground, going for a walk, on a plane, at the airport, or just on a play-date.
Of course, band-aids are good because they cover the wound and stop the bleeding, but for young kids they also have some powerful painkilling properties. Many parents have probably experienced this: a small scrape that magically seems to stop hurting once a band-aid is on it!
Here is what I carry with me in my very basic, first aid pouch:
Band-aids, various sizes
A variety of sizes is nice: some teeny-tiny ones for fingers and toes, while skinned knees often require really big band-aids for full coverage. Using band-aids with your child's favorite cartoon character can sometimes seemingly increase its "painkilling powers"...
Something to clean wounds with
Individually wrapped antiseptic cleansing wipes are perfect for this, but a small bottle of antiseptic spray works too. Carrying an antiseptic wash and some cotton pads is another option.
A small tube of antibiotic cream is handy to have on hand too.
This is mainly needed in spring and summer, and it can really be helpful if your kids get stung or bitten when you're out (especially when you're hiking in the woods). I like the Afterbite brand, and usually carry a tube with me. It really works to help take the sting and itchiness out of insect bites.
I always carry a small bottle of this with me because when you have kids, your hands (and their hands) sometimes end up in some not so clean places... and it is really nice if you can get reasonably clean even when there's no sink or soap nearby.
Tips for air-travel
I bring my band-aid pouch on the plane in my carry-on, but to comply with the regulations for liquids on board, any liquids, including sprays and gels, have to go in a separate plastic bag, and each bottle can be no bigger than 3 oz (100 ml). Wipes of any kind do not count as liquids, which makes them a good option on board.