For example: Those escaping extreme poverty and/or politcal problems.
Welcome to Mexico and most of Central/South America.
To fix the problem of illegal immigration weed need to do two things-
1. Hire as many border patrol agents as it takes to secure the borders.
2. Go after the businesses that rely on the cheap, illegal labor.
I know a lot of people are saying "no one wants the jobs illegals take". I'd say you're wrong. The illegal immigration from Mexico has exploded in the last 25 years. Before that there was illegal immigration, but nothing on the scale it is today (estimates say it's probably over a million a year).
Before all the Mexicans came in to the country, businesses still managed to stay in business. There are people will to take those "crappy" jobs. The businesses may have to pay more to get their toilets scrubbed, but it would still happen.