There is no other way to guarantee enough workers in a rapid aging society such as Japanese one. And of course, without enough workers, the productivity will decrease and the economy will decline. Certainly, there are some bad things about accepting immigrants. But do you want to be a source of cheap labor ?
Immigrants will prevent our population from aging to the point where it causes economic calamity trying to take care of our elderly. Immigrants also become consumers in the economy, stimulating growth and in the long run increasing the number of jobs.
As for "cultural purity" an educational program and a test could help with that, something along the lines of what they use in the Netherlands. And there's not really much to worry about there except for immigrants coming from conservative Muslim and sometimes conservative Christian (think Uganda) countries. When it comes to other cultures there aren't that many points of tension besides language, and America requires knowledge of English for naturalization, so that's taken care of.
The world's most successful and highly regarded countries have the lowest immigration rates. There is a reason. Immigrants bring hands but they also bring hearts and ultimately diffusion between values form so a lot of primarily "white" or "black" or "latino" or "italian" or "eastern" values mesh in melting pots and you get high amounts of collusion that just bring about entirely new generations that veer far from prior concepts. This greatly hinders the ability to organize and manage a society with a distinct column of behavior which in turn greatly hinders the ability to organize and manage a progressive society.
In short because everyone isn't on the same page and there are no clear divides even between age groups anymore you've got no particularly strong ties to any cultural behaviors which chokes out the ability to trace back and grant yourself the skills and abilities needed to perpetuate your own heritage.
It appears that relatively unrestricted immigration leads to a host of issues surrounding law and order UNLESS strict procedures are in place to ensure the immigrant is promising of being a productive member of society without rebelling against existing values of the given culture..
It also sparks further concern surrounding house ownership potential, and competitive job and university placements.
The only reason the left supports immigration is because it means they can import more people to join their class struggle against the right. They can't win the votes of the locals the same way the right can and thus they just want to import new voters instead.
In countries like Sweden where there have been lots of immigrants far-right parties are now gaining popularity as a result. Then there is also the issue of too much immigration harming the labor market causing lower wages and unemployment.
I don't think borders should be completely closed, but immigration should be regulated.