Let me preface that I'm more than conscious this is a delicate topic, and that the average, non-latino American really lacks the understanding of how strongly non-cuban latinos feel about this. Thus the position I described earlier about political suicide and such.
The reality is that the US has a 'humanitarian crisis' type of immigration visas, but that's not what's going on here. If some of these people are desperate, then they should apply for asylum in the US or any other country and simply respect and go through the laws and process. The average latino though, it's very "flexible", shall we say, about that. It's a different idiosyncrasy, but that doesn't make it right. As far as the jobs that Americans really don't want to do and are more of a seasonal type of job, well, let's find a middle ground: temporary worker visas, they can do the job, and take off when they're done, with a good pile of money wherever they go back to.
Let me tell you a story: back in WWII, my grandfather was headed on a ship to New York, after his brother was captured and killed in the War. New York closed down the port and simply said "we're full". Ship refueled, and headed to the 2nd largest American port at the time, Buenos Aires. And that's how my family ended up there. This is WWII we're talking about, and yet the US finally figured that they couldn't take anymore. At some point we're going to have to figure out that we're only going to take in people that can show they're really desperate and that they truly need asylum, going through the proper channels.