Theoretically speaking, I'd be against allowing non-members of a party a vote in choosing that party's leadership or candidate for an election.
But the United States doesn't have real political parties like other countries do.....so one might view the primary system as something similar to the first "round" of an election.
Still, my answer is no. You can "register" with the party if you want to vote in its primaries.
In South Carolina you "register" by voting in the primary. If the parties want truly closed primaries, then they, not the government, should be the one to pay for the expense of running the primary.