As a med student interviewer, I would love to see an applicant with an interest in Biostatistics, and of course this can be demonstrated by completing an MS program, or in other ways (ie research experience). I would be surprised if a medical school adcom looked at this as a negative - if they did not value it, and if you really have an interest in biostats and its associated use in clinical or translational research, then that institution would not be a good fit for you, so screw 'em! 🙂
Some things to consider/questions:
Where did your interest in biostats come from? have you taken biostats classes? used stats for research? I only ask because you might be better served (from a time and $$ perspective) to apply to medical schools that have joint MD/MS programs, and complete your MS in biostats that way. It will probably cost less, take 1 year vs. probably 2 years, and will be more useful for you as you will be able to tailor your course experience based on interests you develop while in medical school, which may be more focused and useful vs. courses taken before medical school.
Biostats knowledge is invaluable in clinical, epidemiological, and translational research, but FYI many institutions have a biostats department that can help investigators address their biostats related questions. With that being said, I have taken biostats courses specifically to ensure that I understand what I am doing in my research, and its been an invaluable skill to have (although I do sometimes still consult our biostats faculty for a second opinion/clarification/etc).
To answer your question - you can apply to medical school during your last (in most cases second) year in an MS in Biostats program, but just make sure to finish the MS before starting medical school.
Finally, please don't complete an MS in biostats for the sole purpose of inflating your medical school application - make sure you have a interest in APPLYING what you have learned in the future as a physician scientist! Also, please take my advice (and really all pre-med advising) with a grain of salt - best of luck!