I went early August last summer and have some opinions that might help.
1. In general, August is more likely to be crowded than June
2. DLR has different Annual Pass (AP) programs, and the AP programs with blackout dates allowed people in 2005 up until June 26, I believe. This means there was a surge of AP people in June trying to hit DLR before the blackout kicked in.
3. Similarly in August, the blackout lifted on Aug 22, I believe. So many AP people with pent up demand came back to DLR.
4. Some people like me care a lot about what rides are operating. FYI, Pirates of the Caribbean (POTC) will be down for refurb until June 22 (
http://www.riderefurbs.com ). If it were me, I would schedule for after June 22, as POTC is a family favorite.
5. June is a month for "grad nights", where high school seniors go to DLR from midnight to 6AM, or something like that. This goes until about mid-June. DLR is open essentially 24 hours during this time, wtih shorts breaks. DLR has to cut back on or reschedule Early Entry programs during this time.
6. Weather is usually cooler in June.
7. Park hours are usually shorter in June. This past summer DLR maxed out park hours starting July 1 and kept them at max until Aug 21. If you have the stamina, the long park hours are nice.
If it were me, I might go in June if it was after June 22. Otherwise, I would go in August if those were my two choices. But in both cases, I would schedule my DLR days to miss the weekends. If I had to be there on a weekend day, I would plan that day for DCA.