Being in the UK, this is not really my problem. And being an island it is easier to control what or who comes in and goes out.
We do have the problem of the EU though, which requires us to have free borders (within certain wide limits) across all EU countries. The problem is that Brits are less inclined to go and live in somewhere like Romania than Romanians are inclined to come here. However, the long term goal of the EU is to make the poorer countries richer and so everyone benefits. It seems to be working in Poland but they have been with us for decades now. Greece on the other hand was a get-rich-quick case and that only ended in abysmal failure.
I agree with TH that controlled immigration is both acceptable and indeed necessary to keep your culture and your economy vibrant and responsive. However, there are limits and I very much sympathise with the problem Texas is having right now. There are two approaches to solve the problem.
Firstly, make illegal immigration much more difficult and risky.
Secondly, make the countries the people are emigrating from wealthier by promoting trade, improving prospects, etc., as much as your own country has the influence to achieve. That will mean fewer people wanting to leave those countries. But this is ok for the EU where you get to control some of what cooks in those countries but much harder when you don't have so much influence.
And I totally get the idea that new immigrants will be voting mostly left-wing politically since it is the left who are are so quick to hand out the benefits. And it's always those poor downtrodden minorities who so much need supporting to stave off the predations of the wicked people who through their own lifetime's hard work have become wealthy. It's short-term and idealistic thinking.