If we stick to economics only, and don't stray into ethical, political or other types of issues, then the argument for or against immigration depends on both the specific economic environment and the point of view of the individuals involved.
While there is not going to be a consensus on this issue, it's beneficial to understand the three main economic issues regarding immigration: supply and demand, productivity and comparative advantage.
Supply and Demand
If supply increases and demand holds steady, then price falls. Therefore increasing the supply of people available in the labor pool, holding all other things equal, should reduce labor costs/employee wages. The phrase labor costs/employee wages describes the different perspectives of the issue, basically employers vs. employees. If you are an employer, labor is a cost. If you are employee, the employer's cost is your wages.
If there is an increase in the labor pool, the additional supply of workers should reduce labor costs/employee wages. But if there was an undersupply, as there was in the tech bubble of the late 1990s, the strong demand increases labor costs/employee wages. In fact, back then it was so strong that there was talk of increasing the number of visas to increase immigration and relieve the inflationary pressures on the labor force.
Unemployment was under 4% and firms were having difficultly hiring workers. This added to their costs and made increasing their output more difficult. So when we think of immigration in terms of supply and demand of labor, whether you find it good or bad really depends on which side of the fence you are on. (To learn more about how the price of products is determined, read Economics Basics: Demand and Supply.)
The output in dollars for each hour of work is crucial as the amount of GDP per person derived from new immigrants may be more or less than the current level. Therefore, while GDP would increase overall as new workers found jobs, the increase may or may not boost the productivity of the work force. Typically, it would not but it would depend on the specific circumstances.
So if the immigrants are high productivity workers, for example tech workers during the tech boom, per capita GDP could increase and overall that would be a good thing. But if the immigrants either don't work or don't increase productivity, the losses would be a drain on GDP per capita.
Another way to think of it is to consider that goods produced using lower cost labor makes them less costly, and frees up resources to produce more of something else. For example, working in the fields to bring in crops is essential. We can't go without food. Native workers would be doing this work unless lower cost immigrants become available. When they do, workers that have been working in the fields can move up to more productive work. This is an increase in productivity that benefits all.
However if immigrants take other positions and displace exiting workers by moving them down into lower wage jobs, then the result may be more profits for the company, but at the existing workers' expense.
Economists use the term comparative advantage to mean the ability of a person or country to produce a particular product or service at a lower cost than another person or country. This concept is usually applied to trade between nations, but it can be applied to immigration. (To learn more, check out What is comparative advantage?)
Take the tech boom as an example. The high tech workers that would have been allowed to immigrate using a new visa program had a competitive advantage over the existing workforce due to their knowledge and ability, not just their lower cost.
The alternative of hiring untrained workers, delaying projects while they go through the training process and paying them along the way is far more costly than simply hiring trained immigrants, even at the same cost. If this had happened, tech wages would have been stabilized or even been reduced depending on the amount of immigration.
If the demand for tech jobs had continued, but without immigration, then existing wages would have continued to rise and there would have been a shift in the workforce toward the more lucrative tech jobs. This would have eventually been good for some workers, but would have increased costs and decreased profits for the firms in the long and the short term. (Loosening labor restrictions has both good and bad effects for a country and its workers. Check out The Economics Of Labor Mobility.)
The Bottom Line
The economic arguments in favor of immigration are that it reduces costs, causing prices to drop and freeing up resources to be used elsewhere. But if you are a displaced worker and forced to accept lower wages or a position that was below your training and abilities, you would clearly be against it. Overriding those two forces is the impact of the economic environment. Today, with unemployment at 25-year highs, immigration is moot. But the issue isn't history; as the economy heats up, those two opposing views will once again start slugging it out.