Reddit mentions of A Pilgrim's Guide to the Camino de Santiago: St. Jean • Roncesvalles • Santiago (Camino Guides)

Depends what your goals are. June is before the peak, so there's a lot of traffic but not to the point where you're racing for berths in albergues. Personally, I prefer solitude and quiet so I found the first week of March ideal - I'd leave at dawn walking quickly and rarely see more than two or three other peregrinos until I stopped for the day.

This past year, about 35,000 pilgrims arrived in Santiago in June, while there were almost 55,000 in August (not all of those on the Camino Frances). One thing I'd recommend if you're concerned about it being too crowded is to plan your pilgrimage to avoid the Brierley, Miam Miam Dodo, and Jakobsweg stages - most pilgrims will stick to those, so the "in between" towns will still have vacancies in the albergues even during busier seasons.

*Edit: I wanted to clarify my references to the various guides - they're all tremendously useful (I carried a stripped down version of the Brierley guide in March) but I'd recommend choosing different towns for your daily rest than they have.