June is hot but not all day, mainly between about 11 to maybe 5, then it cools down to a perfect temperature and with a nice light breeze. Actually the main thing is the sun. If you are in the shade during that time, it's warm, but fine. Restaurants are open, hotels, aside from Los Arcos, are open and people will be walking up and down the malecon in droves by 6 o'clock or so. There are several nice hotels right off the ocean right in town or you can go north a bit where there will actually be more of a beach area. Crown 7 or Hotel Perla if you want to be down town (though there are plenty of others). Costa Baja to the north a bit would be a nice place to stay for a more of a resort experience. If you're going to be out during midday, just bring light, white, airy clothes and hats to block the sun.
I would say it is crazy to stay in La Paz during August and September with hurricane possibilities and definite heat/humidity. But June is fine.