"记录上班时间" can be translated into English as "record the start time of work" or more simply and commonly as "log the start of work hours" or "clock in (to record the start time)".
"Record the start time of work" directly translates the meaning, emphasizing the act of noting down the beginning of the work period.
"Log the start of work hours" is a more colloquial way of saying it, often used in contexts where employees need to keep track of their working hours.
"Clock in" is a very common phrase in English-speaking workplaces, referring to the act of marking one's arrival at work, typically by using a time clock or a digital system to record the start time. Although it doesn't explicitly say "record," the context implies the recording of the start time.
Choose the expression that best fits the context and level of formality you need.