There is an annual cycle at play here. The peak season starts with Semana Santa (Holy Week) just before Easter. Traffic builds from that point until the schools let out, particularly in Europe at the end of May - through mid-June.
After that, things get crazy busy through about mid-September. This is when most students and school personnel have time to do pilgrimage. It is also the annual holiday / vacation season for many people.
To get a good idea of the traffic, look at departures from St. Jean Pied de Port. Probably the single largest tranche of long walkers, on the most popular Camino (Frances) leave from there. There is a sharp drop off in people starting around mid-September. By monitoring this 'pig in the python' you can get a fair idea of when it is 'safe to come out and play...'
The other really large starting wave on the
Camino Frances is from Sarria, at almost any time of the year, as this is the most popular starting place for folks who can or choose to walk the 118 remaining kilometers on the Caminos Frances. I call these minimal distance folks the 'short-walkers.' It is also easily accessible by bus and train from within Spain.
Recently, Tui (Spain)and Valenca (Portugal) on the Camino Portuguese have increased in short-walker departures as well, as the distance is also about 118 - 120 km. This makes one eligible for the Compostela. Valenca is easily accessible from the south in Portugal. Tui is also easily accessible by RENFE train from Spain.
Hope this helps.