Depending on the nation and the era, condemned criminals did in fact pay executioners, typically in an attempt to prevent a botched execution. The popular legend that criminals tipped the executioner is not true, however, and the custom of paying the executioner has waxed and waned historically. More commonly, condemned criminals bribed guards and prison wardens to have access to more comfortable cells and to secure permission to import special food, books, and other diversions to entertain while waiting for the day of execution.
When the guillotine was first introduced, some condemned criminals would pay executioners to sharpen the blade, ensuring a quick and relatively merciful end. Prisoners sentenced to beheading in certain eras in England would also pay their executioners, requesting execution in a single blow. In both of these senses, the payment was more like a bribe than a specific fee for services rendered, as it were. Hangmen, as a general rule, were not paid, and likewise with firing squads.
At times, family members have also received bills for execution, most commonly in the case of military executions in the 18th and 19th centuries. In these instances, the family might be ordered to pay for the hanging rope or bullets used by the firing squad, along with the soldier's uniform. Sending out a bill for execution was designed to act as an extra deterrent to members of the military considering crimes punishable be execution.
In the modern era, it is not customary for condemned prisoners to pay executioners. In fact, some criminals never meet their executioners. In the Western nations which retain the death penalty as punishment, such as the United States, the process is shrouded in anonymity due to social stigma, with executioners typically concealing the nature of their jobs from all but a few close friends and family. In nations where the death penalty is more openly practiced, such as some Asian and Middle Eastern countries, it would still be unusual for criminals to pay executioners.
Executioners historically were usually paid by their clients, for lack of a better term, when their regular rate of pay was generally very low. The idea that some criminals could pay while others could not raises the interesting and horrific specter of classism enduring even at the gallows, because it suggests that if criminals didn't pay executioners, the execution might be less humane.