It did, but not anymore. From May 30, this year:
Gov. Kay Ivey has signed a law that prevent crossover voting in the state. The bill, sponsored by Sen. Tom Whatley, specifically prohibits voters from casting a ballot for one party in a primary and then crossing over to vote in another party's runoff elections.
"It helps the Democrats choose Democratic candidates, it helps the Republicans choose Republican candidates. It just prevents the cross-over voting so you get a pure general election with a Democrat and a Republican," Whatley said.
This is excellent news for Judge Moore and really bad news for McConnell, Strange, et al.
I read the source you posted and it is good news but I will tell you that very few democrats actually voted in their primary. 90% did not vote. McConnell will solicit door to door faking a voter registration drive and pay peoplwe to vote for Strange claiming the Roy Moore is racists. I see it coming like a freight train.