The reason you choose a shorter wavelength "black" light such as 365nm, rather than a longer wavelength near-UV "black" light such as 390nm, is that it is right in the middle of the longwave-UV spectrum (~320nm to 395nm) where it can affect far more fluorescent materials (such as red invisible inks) with greater intensity, and because it produces far less visible light for better contrast.
A 365nm wavelength black light is best achieved with fluorescent black lights because the mercury vapor technology they use already produces wavelengths that can be converted to 365nm relatively efficiently. The best black lights usually use full-sized tubes, but specialized compact versions also exist. LEDs are not as efficient, because of the amount of energy that is required to generate shorter wavelengths with such materials; it's also a reason why shorter wavelength LEDs are expensive and were some of the last LEDs to be developed.
Most people will end up getting some cheap "UV" LEDs from eBay, and they're at 395nm or so. That's fine if you want to fluoresce white or blue colors or maybe even fluorescent yellows or greens. If you want to light up other wavelengths (reds and such), and/or really produce an authentic bright black light appearance (everything but your fluorescent colors appearing black), you should seek out true black lights. LEDs in "true" black light 365nm do exist but are very expensive
compared to regular cheap "UV" longer-wave LEDs. Even small 5mm 365nm LEDs will cost about $1.25 a piece. I bought a 10W one for $110 or so and the price really hasn't gone down even over the past year and a half. Getting that thing to work required a special heatsink, and a special power source that provided 19V and about 800mA. Note that the 10W corresponds to input
power (yes, I was driving it higher than its supposed rating); the output power -- in terms of light -- was only about 500mW with the rest going into heat.
Black lights produce light in the UVA territory. While all UV light can affect your DNA to some extent, you usually won't be exposed to enough of it over long enough durations to make a difference. Do not use UV sources that produce light at wavelengths 320nm or below. That is moving into UVB territory, and that is very harmful to your body. UV wavelengths even shorter than UVB are UVC. Germicidal fluorescent bulbs produce UVC at ~260nm, which severely damages DNA -- consequently making them work very well for sterilizing purposes.
Note also that lumens are not a useful measure of power with respect to UV LEDs because lumens are based on the eye's ability to perceive light, and also because using UV LEDs for black lights is mostly for the fluorescence they cause, not the perceived (or rather imperceivable
UV output). You should, however, take into account how bright a color will look
for a given output power. For example, those blue lasers (445nm) you see on the playa are being run at around 2W output power, and yet will appear less bright than a green laser (532nm) running at less than half that power. At night, the worst color as far as brightness is concerned is red. Green is usually always a good choice if you want to stand out.