The real truth is that the High Sierra is closed until June 15th. Nobody goes. Then on June 15th, everyone is allowed in, but they have to demonstrate their ice axe and crampon skills first. 
It varies by the snow level each year, but i think the best bet is to go early or late. Generally mid-late June and early July is going to be very mosquito-y, and the river crossings will be deeper and faster. August is drier and easier. But early June or even May (depending on the snow year) is even more stunningly beautiful. It just is harder work, dealing with snow and occasional navigation. Ice axes or even crampons might help some people, but there are only 3 passes steep enough to warrant using them: Forrester, Glen, and Mather. If you want to not carry the extra gear and weight, try to time these passes for mid-afternoon. Then the snow is softer, and it’s generally safe to plod through, as opposed to morning when it’s iced over and scary as hell.
Of course, YMMV, and if you go early season without an axe, you definitely have to be be willing to look at a snow slope, assess it, and turn around if you feel it’s not safe or comfortable for you. There are plenty of exit trails (that are equally beautiful) if you find you can’t get past a pass safely.
markv