The Unofficial Guide to WDW website is predicting slightly smaller crowds for the first week of June, though both weeks will certainly be crowded.
However, note that the first week in June will overlap with Gay Days. If that's a concern for your family, check out the Gay Days thread for more information on that. I've never been, but from what I understand, they make plans to attend certain parks on certain days, so you may be able to avoid the bulk of the Gay Days crowds by attending other parks on those days.
-p.b.