Yep, you would prefer to draft as close to opening day as possible, but I wouldn't (and don't) avoid leagues just because they draft early. The fact is, everyone in the league is drafting early, so you all face the same risk. Sure, it sucks if your guy gets injured, but when you have a 12 team league, odds are good that it won't be you having to deal with it.
Also, there is a lot of positive news that comes out in training camps and preseason as well. If you stay up on the news, you can be the guy that takes advantage of the positive stuff through sneaky WW pickups, so you would actually end up benefitting from the early draft.