My university requires that I take attendance. How should I treat students who arrive late to class? I might soon teach at a university which requires that I record attendance of students, as follows:
Attendance is a requirement for the course. A person with at least 75% of attendance is eligible to take final exams. A person with an attendance between 65% and 75% is required to submit a medical certificate. A person with less than 65% of attendance is not allowed to write the exam and has to do recourse. 

As the university has no explicit rules on how attendance should be recorded, I am asking about this.

I studied many courses, and the behaviour of faculty in this particular aspect varies from one another. The following are the behaviours of faculties I observed mostly: 


Allowing the latecomer into the class at any time and giving attendance to the latecomer.
Allowing the latecomer into the class at any time, asking a reason for being late, and then giving attendance to the latecomer.
Allowing the latecomer into the class, but giving attendance based on the reason the latecomer gives for being late.
Disallowing the latecomer by closing doors or with strict instruction.


So, I came to the impression that it totally depends on the particular faculty. Mostly I observed 1, 2 and 4. I observed 3 very rarely.

What is the recommended and proper behaviour towards latecomers?
This is an answer to a previous version of the question, where it was not clear that recording attendance was mandatory at the asker’s university. Therefore my answer does not cover the specific situation of the OP, but may still help others in a similar situation.

0)  Allow all students in class and do not monitor attendance.

They are adults and they come to learn. As long as they do not disrupt the class they are free to come and go as they please. I do not see which purpose the monitoring of attendance serves. It sends the wrong signal and focuses students on signing the register sheet rather than on the objective of the class.

I do, however, start each class by thanking students who attend and come in time. Latecomers will miss this part.