I've crossed that part about 12 times in my life in June.
Get there around 4 in the morning. Depending on how far north you are going (we go north of Red Lake) you want to hit I believe it's 502 south into Dryden, during the light. We hit it at night once and we saw about 45 moose. It's 2 hours of driving around curves, right in the middle of the woods. No places to stop, only pulp check stops, and adding in the moose makes it about the most stressful 2 hour drive you'll ever make. If you're driving through that, the border will be the least of your problems.