Why Does Government Fund Party Primary Elections???

Darned good question raised here at this link generating comments.

ScottW kicks it off with this excellent logical comment that draw following comments in the thread.

"I confess to knowing nothing about election laws, but there is something that seems unconstitutional about limiting someone’s vote to a party affiliation in elections that are funded by the taxpayers. Why shouldn’t I have a first amendment right to vote for who I want to vote for in government funded elections? Limiting my vote because of party affiliation is a content based restriction on my right of free speech.
To me the nexus is government action combined with restrictions on free speech/expression. And if corporate money is considered speech, my vote should certainly be considered speech. Telling me I have to vote for selected candidates restricts that right.
Could they get away with the same thing in the general election–or would it be unconstitutional? I know–as the judge used to ask me–do you have anything other than the constitution to support your argument?"

Political parties look like private parties.  Especially when someone must be a registered member to vote.  OK, if that is the case then government might pay for the "voting booth" infrastructure necessary for all to vote in primaries only if all voters can vote for all candidates without restrictions. 

Info on government paying for conventions

http://uspolitics.about.com/od/Political-Conventions/a/Taxpayers-On-The-Hook-For-Political-Conventions.htm