Safety and quality of low-price imported beef sold in US supermarkets

It should be good. They're not excellent cuts, but great for marinading. Mexican beef has always been a cheap, but easy to work with choice for me.

Exporting meat into the United States is not a simple or easy matter.

From the United States Department of Agriculture (USDA) Food Safety and Inspection Service (FSIS):

Checklist for Importing Meat, Poultry and Processed Egg Products

This checklist is provided as an overview of the steps needed to be taken when you want to import meat, poultry, or processed egg products to the United States.

1. Products must originate from certified countries and establishments eligible to export to the United States.
2. The Animal and Plant Health Inspection Services (APHIS) restricts some products from entering the United States because of animal disease conditions in the country of origin. For information on restrictions related to animal diseases and information about APHIS, contact the APHIS Veterinary Services, National Center for Import and Export.
3. Countries and establishments become eligible following an equivalence determination process by FSIS.
4. Imported products must meet the same labeling requirements as domestically-produced products.
5. After filing the necessary forms for U.S. Customs and Border Protection, and meeting animal disease requirements of APHIS, all imported meat, poultry and processed egg products must be presented for inspection by FSIS at an official import establishment.

Mexico is one of 33 countries that can legally export meat to the US. In Mexico there are 74 companies authorized to export to the US (As of May 3, 2017). USDA Source

Also from the USDA Website:

So, in the United States, beef from Mexico has been determined by the US Government to be as safe as domestic beef.

The answer provided by @Jolenealaska is comprehensive on the question of 'safety'. As one who has raised beef cattle I will address the 'quality' question you raise.

In the US and Mexico there are a wide variety of beef cattle breeds However, the Mexican ranchers 'lean toward' hybrids that are Brahma crosses because these are heartier (physically) breeds that can more easily withstand the climate (as Brahman). Often you will find that Mexican beef is tougher and introduces a 'gamey' quality to the flavor of the meat.

IMHO while you can rely on Mexican Beef to be 'Safe' (by USDA standards) it is not necessarily of 'high quality' for your table.

For a more detailed review of the Mexican Cattle Industry see http://www.gbcbiotech.com/bovinos/english/bovinos.html

The other answers and links here cover the safety issue sufficiently for me. As to quality, here is my offering.

I have found a small market near me that sells beef quite a bit less expensive than the chain markets, or Costco, and Sam's Club. I have been getting very nice looking porterhouse steaks for $2.99-3.99 lb (sold as family pack 2-3 per package). I suspected they were Mexican beef, but I didn't ask till today.

I've been buying meat for many, many, years, and I'm a pretty good judge of what good meat looks like. These steaks are usually pretty well marbled (you have to look closely, it varies) , nice color, and when cooked (I like no more than medium rare) are as tender as any other steak with similar appearance.

The flavor is quite acceptable to me, and I don't consider it "off" in any way.

I'm going to continue buying these Mexican steaks.

It is safe as U.S.D.A inspected. I do wonder if it is from tougher breeds of beef? Beef can be aged at 27f to 29f in shipping. To age it. Make it tender. As bacteria works on it at that temp. Tenderizes it. So some what different than American grain fed beef. Little different texture & taste. But safe. I am more familiar with Australian beef that is shipped that way to Europe & Asia.

I live in Mexico and the beef here is generally better than what I had in the US. Tenderloin for unknown reasons is one of the cheaper cuts. I've been buy the entire tenderloin for $20~30 (4~6 lbs) and cutting it myself to use as fillet mignon, Chateaubriand, birria, etc. In the last few years the stores have been using US cuts like rib eye, New York, Porterhouse, t-bone, etc. Never had any issues with bad meat from known stores.