I don't think immigrants make up the basis of the Swiss economy. Looking at their demographic data[1], it was pretty strongly European dominated for a very long time, and is still ~80% European.
When people talk about immigrants in this context, I don't think they mean people from the US, but lower socioeconomic asylum-seekers and refugees etc. from the middle-east.
It definitely seems their economy was built on European labor, which I believe the vast majority of European countries were.
It depends somewhat on the specific campaign but generally in right-leaning Swiss politics, "immigrant" includes Europeans as well. I think that's definitely the case in this initiative since the text of the initiative requires ending freedom of movement with the EU.
But, it does seem like a terrible place to live. I fear for the immigrants who form the basis of the economy.