State law determines the rules of primaries and not all states have primaries. I am not aware of a single state that would allow you to participate in both Republican and Democratic primaries or caucuses during the same election.
Some states have open primaries where you don't have to affiliate yourself beforehand and get to choose which one to participate in, but you can't participate in the other without risking being charged with a crime. Generally if there's a runoff in one of the primaries you can only participate in the runoff for the one that you participated in.
Others have a semi-closed primary (or semi-open if you prefer), where only unaffiliated (independents) get to choose, registered Republicans and Democrats can only participate in their respective primaries.
Still other states don't have a primary system at all but a caucus system in order to choose their party nominees, and those rules vary from state to state. Iowa is the first caucus in the election season so it winds up getting the most attention nationally, but other states that caucus in some form are Alaska, Colorado, Hawaii, Kansas, Louisiana, Maine, Minnesota, Nebraska, Nevada, North Dakota, Washington, and Wyoming. The Texas Democratic Party had a complicated system that included both a primary and a caucus, but the national party got rid of it in 2015. In 2008, even though Hillary Clinton won the popular vote, Barack Obama won the caucus and received more delegates from Texas in total than Clinton.
If so, then would a movement to encourage this help to reduce polarization by enabling a candidate to win nomination with more centrist policies?
Maybe, but with primary turnout in 2016 at around 29%, I would think it just as likely partisans from both sides would attempt the opposite by voting for the more extreme candidate in the opposing primary hoping it would make it easier for their preferred candidate to get elected in the general election. We can't predict the future impact or how people will choose to vote.
And while the merits of each system can be debated, ultimately the government can't force the parties to choose who will represent them in a particular way, it's the parties who have to choose. Viable "third parties" could upset the current status quo, but since the two main parties have such high incumbency and structural advantages you often see those candidates who might not actually be party members aligning themselves with one of the two in order to gain their support and avoid becoming nothing more than a spoiler. In my mind further polarization makes the likeliness of third party genesis higher as those left in the middle feel left behind, but the two major parties have been pretty adept at staying in power for quite a long time.