The original version of the "temporary worker program" was passed in 1952, and it's been continually revised over the years, in the direction of expanding it mostly. Some history here. The current version of the program dates to 1990, in legislation that was passed by a Democratic congress and signed by a Republican president.
But history aside, is there a meaningful partisan divide on this issue? My impression is that when it comes to actual legislative action, both parties have been mostly in favor of the program. There is opposition in both parties as well, in the Republicans mainly from the anti-immigration faction, and in the Democrats mainly from the union faction. But not enough opposition to do much about it.