The derogatory noun wetback denotes an illegal immigrant who crossed the Rio Grande from Mexico to the USA.

This term seems to have entered the congressional record and the national political lexicon in 1920.

These are the earliest occurrences of wetback that I have found:

1-: From the Fort Worth Record (Fort Worth, Texas) of Friday 13^th February 1920—wetback is used attributively:

Washington, Feb. 12.—F. W. Berkshire, immigration official at El Paso […], appeared before the house immigration committee today to testify as to the necessity of an adequate control along the border to keep undesirables from entering Texas and Arizona from Mexico.
[…]
Berkshire said he would not attempt to estimate the number of “wetback” Mexicans who have entered the United States by swimming the Rio Grande, but doubted if there has been 100,000 as has been estimated.
He also revealed that there is a difference of opinion as to the entry of Mexicans as farm laborers. He said the need for laborers is great, and yet many people feel that bringing them in as farm laborers will eventually build up a great Mexican population in Texas.

2-: From the Brownsville Herald (Brownsville, Texas) of Saturday 21^st February 1920:

HOW “WETBACKS” ARE DISCOVERED ALONG BORDER
Witness Before Immigration Committee Tells of How Labor is Imported

Washington, D. C.—Congress was introduced to a brand new term when a witness before the house immigration committee, discussing the Hudspeth resolution to admit Mexican labor for agricultural purposes ¹, told how the Texas land owners procured “wet backs.” These are Mexicans brought across the border at points where there are no immigration inspectors, without payment of the $8 head tax or standing the literacy test. They are supposed to have swam the Rio Grande, and their backs are still wet when they reach the plantation.

¹ In 1920, Claude Benton Hudspeth (1877-1941), U.S. Representative from Texas, introduced a resolution to allow illiterate Mexican labourers to be temporarily admitted to the USA.

The noun wetback has become a generic term for any illegal immigrant who entered a foreign country by swimming—as in the following from The Guardian (London and Manchester, England) of Friday 8^th June 1979:

The Foreign Secretary, Lord Carrington, has summoned home the Governor of Hong Kong, Sir Murray MacLehose, for urgent consultations on the crisis confronting the colony over the influx of boat people from Vietnam, and the even larger numbers coming from China.
The topic was also taken up by Lord Carrington with the Chinese and Soviet ambassadors in separate calls at the Foreign Office yesterday.
[…]
What Lord Carrington is seeking from China is an agreed formula on the outflow of illegal migrants from South china who are getting into Hong Kong by junk or sampan, usually swimming the last part of the trip. The total of Chinese “wetbacks” intercepted by the Hong Kong authorities in the first week of June alone came to 3,722, and the total for the year so far is 37,000.
The authorities estimate that for every “wetback” intercepted, another three get through. On top of that there is the flow of legal migrants at the Lowu crossing point, which has shot up since last year and has already reached 50,000.

In a sardonic twist on the much-resented U.S. noun wetback, some Mexicans took to calling espaldas secas ² (translating as dry backs) the U.S. citizens working in Mexico as a result of the North American Free Trade Agreement ³.

² The Spanish feminine noun espalda translates as back (i.e., the rear surface of the human body from the shoulders to the hips); the Spanish adjective seco, feminine seca, translates as dry.
³ The North American Free Trade Agreement (NAFTA) was an agreement signed by Canada, Mexico and the United States that created a trilateral trade bloc in North America; it came into force on Saturday 1^st January 1994.

The earliest occurrences of dryback that I have found are from U.S. ‘drybacks’ come seeking work: Mexico under NAFTA, by Andres Oppenheimer, Herald Staff Writer, published in The Miami Herald (Miami, Florida) of Sunday 3^rd July 1994:

Mexico City—When Mexicans talk about illegal aliens, they aren’t necessarily referring to the millions of Mexicans who have slipped into the United States without proper immigration documents in search of a better life.
Increasingly, they’re speaking of the growing numbers of U.S. citizens who have come to Mexico in search of jobs, some of them lured by the publicity surrounding the North American Free Trade Agreement (NAFTA), and who stay here without residency permits.
They are known as espaldas secas—drybacks.
Unlike the “wetbacks”—a derogatory term for Mexicans who often sneak across the Rio Grande and into the United States—most of the U.S. citizens arrive by car or in comfortable plane seats, and are automatically granted tourist visas.
Mexican officials estimate there are about 200,000 U.S. citizens are [sic] now living illegally in Mexico, in addition to the 150,000 registered with Mexico’s National Migration Institute as permanent residents. Another two million visit every year as tourists and return home.
While many of the U.S. immigrants are California and Texas retirees lured by the cheaper cost of living here, growing numbers are young, well-educated, and come from all corners of the United States. Many are looking for foreign business experience to give an international flavor to their resumes.
[…]
In recent months, as Mexico has had to revise its immigration laws rules to comply with NAFTA regulations demanding freer access for businessmen and professionals, Mexican officials have begun paying increased attention to the “drybacks.” […]
[…]
Mexican officials stress that, under new regulations generated by NAFTA, Mexico is making it easier for U.S. businessmen to work here. Starting this month, those coming here to explore business opportunities will be automatically given a 30-day business visa, which should help them cut the red tape if they decided to apply later for a full-fledged business visa.
It may be a hard sell, however, U.S. expatriates say.
Many undocumented Americans prefer to continue working in Mexico with their tourist visas, for fear that a request for permanent visas could be turned down and attract the attention of Mexico’s immigration service.
Mexican regulations require, for instance, that American retirees moving here have an income of about $2,000 a month, and that businessmen prove their association with an established U.S. or Mexican firm.
Independent professionals or entrepreneurs must show that they are not taking away jobs from equally qualified Mexicans.