Relations between the US and Mexico were pretty bad as of January 1917, when the Zimmermann Telegram was sent. Mexico was in the middle of a decade-long series of civil wars (the Mexican Revolution), in which the US had intervened repeatedly, culminating in two separate large-scale military incursions into Mexico.

The first major US intervention in the Mexican Revolution occurred in Feb 1913, when the US Ambassador to Mexico (Henry Lane Wilson) gave critical planning and diplomatic support for General Huerta's coup against the elected President Madero. Wilson was also widely suspected of helping to both arrange and cover up the assassination of Madero that completed the coup, although I don't think those accusations were every proven.

The following month, the newly-elected US President Woodrow Wilson (no relation) dismissed Ambassador Wilson and replaced him. Meanwhile, multiple rebellions had broken out against Huerta, and President Wilson imposed an arms embargo on Mexico. Several incidents occurred relating to American efforts to enforce the embargo on Mexican soil, which the US escalated by invading and occupying the Mexican port of Veracruz in April-November 1914. The occupation was partly retaliation for the arrest of some American sailors by local Mexican authorities in Tampico, and partly an attempt to intercept a German merchant vessel carrying an (American) arms shipment to Mexico (in April, several months before WW1 broke out).

The Huerta government was defeated in July 1914, and another round of civil war broke out between the more moderate revolutionary faction (the Constitutionalists, who were strongly nationalistic but otherwise fairly liberal: they were the eventual victors of the Mexican Revolution and the forerunners of the modern PRI political party) and several of the more radical factions (most notable those led by Pancho Villa and Emiliano Zapata). The Constitutionalists had quite a bit of military success from the start of this phase of the Revolution, and were generally supported by the United States (diplomatically, by allowing them to purchase American military supplies, and by allowing Constitutionalist military units to travel on American rail lines).

Pancho Villa, whose base of support was in the North near the American border, launched a series of raids across the American border in retaliation for American support to the Constitutionalists: one against Nogales, Arizona in November 1915, and another against Columbus, New Mexico in March 1916. In between these raids, Villa's forces also arrested and summarily executed 16 American mining engineers in Mexico.

And that leads us to the second and larger American military incursion into Mexico, know as the Punitive Expedition or the Pancho Villa Expedition. About 10,000 American soldiers (pretty much the largest force the then-tiny peacetime US Army was capable of fielding) invaded the region of Northern Mexico in which the Villista rebels were operating. The expedition was commanded by General John Pershing, who would later be the commander of the American Expeditionary Force in France after American joined the World War. It lasted March 1916 through February 1917, and had mixed success: the expedition fought several skirmishes against Villa's forces and generally won, but it failed in its primary objective of killing or capturing Villa, who evaded American pursuit and would eventually be assassinated in 1923.

The Constitutionalist President Venustiano Carranza did give at least passing consideration to accepting the German offer of alliance: despite American support in the post-Huerta phase of the Mexican Revolution, there was a great deal of resentment towards the US for America's role in the Huerta coup, and for the occupation of Veracruz and the Punitive Expedition (both of which were overt acts of war against Mexico, despite being directed at Huerta's and Villa's factions respectively, not against Carranza). The Constitutionalists also had ideological reasons to resent the US: remember that they were a strongly nationalist party, and one of their big issues was perceived exploitation by American mining and oil companies operating in Mexico.

Carranza declined the offer after submitting the question to his senior military commanders for consideration, on the grounds that Mexico was in no condition to fight a full-scale war against the US even with German support (note that Carranza's government was still fighting both Villista and Zapatista guerrillas at this time), and that Germany's ability to support Mexico effectively was dubious, as Germany was then under an increasingly-tight British blockade.