I saw a reaction between hydrogen and chlorine ignite in the presence of UV light. I was wondering if the same thing would happen with something like near-ultraviolet light (around 400 nm). What kinds of UV light (if any) would start a reaction of stoichiometric amounts of hydrogen and oxygen gas in something like a balloon?
Hydrogen/chlorine needs only visible light to ignite. You can use a normal security light to do this because chlorine can be dissociated by visible light, $\lt 490$ nm which is of a bluish/greenish colour. The reaction is a radical chain reaction initiated by chlorine atoms.
A security light is suitable, they are cheap and bright and can produce a lot of radicals and thus an explosion. If you put stoichiometric amounts into a large plastic lemonade bottle with a rubber bung in the top it makes a good demo. But make sure to use a transparent screen between you and any audience as the bottle can sometimes shatter. The cork usually hits the ceiling and the bang is loud :)
If you want to do the same with hydrogen/oxygen, which is also a radical chain reaction, you will definitely need UV light as the dissociation energy of both is now far larger, oxygen $\lt 244$ nm, hydrogen $\lt 276$ nm. It probably won't work so well in a balloon as the rubber will absorb a lot of UV. (Also, UV light of this wavelength is dangerous to the eyes and skin).