The Hoechst 33342 dye is similar to DAPI in that both are UV-excited, minor groove-binding, and emit signals proportional to total DNA content. Both are maximally excited around 355 nm and emit around 460 nm. A UV light source is required, which may harm the cell. However, this is only a risk with Hoechst, as DAPI requires the cells to be fixed and/or permeabilized, which is incompatible with life. DAPI is more stable, but Hoechst is brighter. Both are subject to photo-bleaching after long exposure.
There is another newer DNA dye called DRAQ5 that makes up for all of the issues I mentioned earlier - it is bright, quite photo-stable, can be used in live cell imaging, and its excitation and emission wavelengths are in the far red end of the spectrum, meaning that no UV is needed. I found a good comparison chart at < if you're interested. I have no connection to them, by the way, I've just used DRAQ5 a lot in the past.