Gammar

1. Do people have a day off on New Year's Eve? (Simple present tense)
and
2. Did people have a day off on New Year's Eve? (Simple past tense)

are both grammatically correct!

But,

Does people have a day off on New Year's Eve?

is wrong in grammer as "people" is a plural meaning noun.

Compared with,

3. Have people a day off on New Year's Eve?
and
4. Had people a day off on New Year's Eve?

are both correct in grammer, but what is the difference?
The answer is 1 and 2 are more common in English and "do" or "did" can make the sentence more stressful.

Do people have a day off on New Year's Eve? (Simple present tense)

The Correct Answer is :

Do people have a day off on New Year's Eve?

( 1 )我們用 Simple Present Tense來表達真相和實況﹗

( 2 )PEOPLE在這裡表示 "眾數" ﹗