What the historical record shows

In a letter addressed to the Cooper Union community on November 1, 2011, its President, Dr. Jamshed Bharucha, wrote: "It is important to note that in the early years (approximately the first forty years) tuition was charged at Cooper Union."

The following is based on the first annual report of the Cooper Union, available online as a two-part document:

The report identifies the following 8 Classes/Departments: 
1. Mathematics: 164 pupils
2. Mechanical Philosophy: 105 pupils
3. Chemical Class: 280 pupils
4. Architectural Drawing: 111 pupils
5. Mechanical Drawing: 104 pupils
6. Freehand Drawing: 102 pupils
7. Vocal Music: 380 pupils
8. School of Design for Women: 120 pupils

These together account for a total of 1366 seats. (It is unclear if some of the pupils might have attended multiple classes).

The 8th one, the school of design for women,  was originally a separate private school, and incorporated into the Cooper Union by adoption of the following special by-law: 

"The Trustees shall establish and maintain a school for the instruction of respectable females in the art of design, and in such other branches of knowledge as in their judgement will tend to the elevation and employment of female labor. The instruction afforded in this school shall be given without charge, but the regulations may provide for the admission of amateur pupils for pay, so long as industrial pupils are not thereby excluded. All money received from such amateur pupils will be required to aid in the instruction of the school, so far as the director thereof may require." 

The report goes on to say:

"It will be perceived that in this school there is a departure from the invariable rule in the other department of the Union, that the instruction shall in all cases be entirely gratuitous.

The Trustees were at first opposed to this deviation, but it was represented by the benevolent and enlightened ladies, who had established and maintained the school up to the time of its incoproration with the Union, that its character and usefulness would be impaired, if the wealthy and refined were entirely excluded from it; that the presence of ladies of leisure and refined tastes tended to raise the standard of art, and to give to the friendless associations of value in reference to their future careers. The Trustees, yielding to this argument, have  limited the number of amateur pupils to one-tenth of the total number instructed. Of 120 pupils in the school, only 12 make payment; and 108 are instructed gratuitously... "

So, the Trustees then made sure that overall only an extremely small minority (12/1366, less than 1%) were paying tuition, and they made absolutely sure that even this small minority was such that it did not need the classes to make a living. The report explicitly says that the invariable rule in everything other than the exceptional case of school of design was that the instruction in "all cases" should be gratuitous.

The distinction made between industrial and amateur students is primarily the following: industrial students needed the classes to help them earn a living, while the amateurs were people of independent means pursuing these courses as a leisure activity. Charging tuition for students that are getting their education for the purpose of building their livelihood would clearly go against the founding principles of Cooper Union.