Apparently, this is not just some crazy idea that Greyhound came up on their own, but it is in response to a lawsuit brought against another bus line that ended up pleading guilty to immigrant smuggling.
Of course immigrant groups are pretty pissed about the policy calling it discrimination against latinos. However, it has been in place for the last three years, and there has been no news about Greyhound refusing to serve customers. It is also somewhat hypocritical of immigrant groups since many have come out against legislation requiring identity documents, making it harder to identify those who are legally here and those who are not.*
The US is a huge country. Just getting across the border is often just the first step in the immigrants journey. They need to get to their destination somehow and busses have to be a popular way to get around. Take a look at the news story linked below and read a summary of the policy. It clearly states what the ticket agents are to look for and it does not appear to discriminate at all and appears to target those being moved by smugglers. Smugglers are not the most upstanding of characters and there is even the possible involvement of organized crime. It makes sense to make it difficult for illegal immigrants to move from the border area. This way it is easier to find them. (Try to remember that they are here illegally)
Greyhound cautions employees to screen for illegal immigrants - Contracostatimes.com
Fighting Illegal Immigration is easier than anyone is willing to admit - Fred Fry International
*The US Government is also guilty in this matter. It took over three years for my wife to get her Green Card "In-hand." This is not an exception, but more so the norm. I NEVER want to have to deal with another employee of BCIS. It is by far the worst US Government Agency. But I will save my experience with them for posting another day.