The GOP has taken a very anti-immigrant stance. According to what they say they want, the borders should be shut to any newcomers. But as can be seen from this graph that shows how the US population would change under various assumptions about the level of immigration, that would not be a good thing.
Immigration is essential to the long-term health of the country, because otherwise people 65 years or older will outnumber children under 18 by 2029, putting stress on medical care and other services.
What the xenophobes are likely most scared about is the growth of the Hispanic population, expected by 2060 to make up 26.9% of the country (currently it is 19.1%) while the non-Hispanic white population, currently making up around 58.9%, will begin to decline in 2045 and may drop to 44.9% by 2060.
One suspects that if the influx of immigrants were from (say) Scandinavian countries, they would be welcomed.