That is a difficult question to answer simply. Technically speaking, they do on the southern border. The State of Chiapas has the “Policia Estatal Fronteriza” (State Border Police) between Mexico and Guatemala. See the link below. http://fusion.kinja.com/as-mexic...
They are poorly paid and as they see it their job is to stop narcotics shipments from crossing the border. They are usually amenable to accepting a small bribe “la mordida” to allow people to pass without papers. Illegal Central Americans are preyed upon all of the way across Mexico to the U. S. border. There is no real attempt to stop them as it is known they don’t want to stay in Mexico but are simply transiting the country on the way to the U.S.
The Northern border is patrolled by the Mexican Army but Mexican Customs mans the Ports of Entry on the border. At one time the Army manned the Pors of entry but corruption was so prevalent, it was estimated that less than 50% of the duty collected at the ports of entry was being sent to Mexico, D.F. At this time Customs replaced the Army at the Ports. Since I worked for the U.S. State Department in Mexico in 2000, the Cartels have become much more powerful and control many units of the Army and police. With Army units actually providing escorts to drug convoys crossing into the U.S. You have to understand that much of Mexico is under Cartel control and the people live in terror.
See the Dept of State travel warning, link below.
https://travel.state.gov/content...