Ridden the Alpe several times. If you think this is something that you'd like to do - if you're inspired, then by all means make sure you do it! Make a plan! It is certainly iconic within the cycling world. There are obviously harder climbs, but the Alpe is.. the Alpe!
If you seek a busy, social experience, go during the high season, but be aware it can get very hot, and the road very busy. If you'd like to experience it more on your own terms, try to do it later, in September or even early October - my first climb was early October, cool and damp, but very little traffic. Take care on the descent, eh? Cheers!