We don't really know
Let's list the information we have:
- According to this answer, Hogwarts could have a few hundred students. Let's say they have 600, as JKR has said at one point. If we assume that those students are evenly distributed across seven years, then each year should have about 85 students.
- Only 17-year-olds could compete in the Triwizard Tournament when Harry participated, limiting it to students in their 6th or 7th years at Hogwarts.
- About a dozen students arrived from Beauxbatons:
Harry, whose attention had been focused completely upon Madame Maxime, now noticed that about a dozen boys and girls, all, by the look of them, in their late teens, had emerged from the carriage and were now standing behind Madame Maxime.
—Harry Potter and the Goblet of Fire
- The number of Durmstrang students is the same:
Harry nodded, hesitated for a moment, then walked slowly toward Mr. Crouch, who did not look at him, but continued to talk to a nearby tree.
“. . . and when you’ve done that, Weatherby, send an owl to Dumbledore confirming the number of Durmstrang students who will be attending the tournament, Karkaroff has just sent word there will be twelve. . . .”
“Mr. Crouch?” said Harry cautiously.
“. . . and then send another owl to Madame Maxime, because she might want to up the number of students she’s bringing, now Karkaroff’s made it a round dozen . . . do that, Weatherby, will you? Will you? Will . . .."
—Harry Potter and the Goblet of Fire
It would make sense for a school to bring all their eligible students, and have them all place their names in the Goblet. Note that Hogwarts, for example, did not hold any eliminations among its own students, but rather anyone who qualified could place their name in the Goblet. The Goblet of Fire, after all, is an impartial judge, and presumably very good at determining which students are most suitable:
Tomorrow night, Halloween, the goblet will return the names of the three it has judged most worthy to represent their schools.
—Harry Potter and the Goblet of Fire
However, Beauxbatons and Durmstrang apparently decided against this strategy:
"The heads of Beauxbatons and Durmstrang will be arriving with their short-listed contenders in October, and the selection of the three champions will take place at Halloween."
—Harry Potter and the Goblet of Fire
Therefore, a lower limit can be determined by assuming that Beauxbatons students constitute the entirety of their school's last year, and about half its sixth year (and similarly for Durmstrang, of course).
Unfortunately, this leads to the conclusion that Beauxbatons has only 8 students per year, giving a total enrollment of 56. Given that France had approximately the same population as Britain in 1994, it seems exceedingly implausible that they would have only a fraction of the students. We are left with a few possibilities.
- More wizards and witches homeschool in France.
- France has far fewer wizards than Britain for historical reasons.
- Students from France go to other wizarding schools.
In any case, 56 serves as a lower bound on the number of students Beauxbatons is likely to have.
Durmstrang has the same limit.
In the films, both schools seem to have about a dozen students.
There are at least 15 for Beauxbatons, which makes the lower limit for school size in the films 70 for Beauxbatons (a bit more reasonable, particularly considering that there may be a separate boys school, which would bring the total up to 140).
There also are at least 11 Durmstrang students, giving a minimum school size of 50 or so in the films.