The first thing to know is your state laws. For example, my state says that a landlord can collect a full month of rent even if the tenant leaves part way through. Some states have laws that say if a tenant alerts their landlord after the first of the month, no matter when they move out they must pay the rest of the current month along with the next. While most landlords won't enforce this, it is a possibility depending on your relationship with the tenant and what state they left your rental property in.
If you are the landlord of the apartment that someone is moving into in the middle of the month, you may also have some issues. The main thing to consider is how you want your tenant to pay the rent. Some landlords request the half month (or whatever period of time exists before the next month begins) and the full next month up front, others simply ask for a month at a time and get the check in the middle of the month. Others still will break it down into future monthly payments until the amount has been met for that little bit of time before they drop down the payments to the agreed upon amount. At the end of the day, the decision is yours but you need to be sure that you keep tabs on how it's going to work.
There is nothing wrong with moving in the middle of the month, but it may take a little extra work on both the tenant and landlord's part. Remember to read up on your state laws to know what your rights as a landlord are when it comes to a resident moving in the middle of the month and how you want to handle it.