Assuming you have everything properly configured, starting mirroring simply means the log rows for transactions not already at the mirror will be sent to the mirror database; once that has occurred the only further overhead would be writing to the mirror database. If the mirror is not very far behind the principal, then this step is generally very quick.
If you have mirroring configured in Synchronous (high safety) mode, that overhead might be enough to show delays in commits. Generally, transactions would be slightly slower since data must be written to both the primary log, and the mirror log before control is returned to the user.
If you have a maintenance window, it would be preferable to start mirroring during that window.