There several dimensions of "space" to consider when deciding stair placement. To understand the space needed you have to thing in 3D.
How much space will be determined to an extent by the measurement from the finished floor of the main floor to the finished basement floor. However there are a few other considerations also.
One "headache" will be enough headroom. The opening to be cut into the main floor has to be long enough so that the minimum headroom in all parts of the stairway will not be less than 80" measured vertically from a sloped line connecting the tread nosing (the rounded front of the tread) or landing to the lowest part of your ceiling line in the basement.
Some times you can solve space problems by adding a landing and turning the stairs in a l-shape or an U shape, sometimes you're just eating up other space. Another space consideration is the width. Most codes call for a minimum clear width from finished wall to finished wall of 36". If they are only service stairs that will be fine. If the basement is going to be finished or you have to move furniture or appliances up and down, I would recommend 42" minimum. Remember to add to the clear width the thickness of wall framing and the finish material of the walls.
Your riser height will determined by the measurement from the finished floor of the main floor to the finished basement floor. Keep in mind that most codes determine that maximum riser height should be 73?4 inches. I like a rise of around 7" for a comfortable stairway when possible.
Tread depth will determine the overall length of the stairs from the riser of the bottom step to topmost riser. Code says that minimum depth of tread, where you step, is no less than 10". For service steps that may work, I try for a depth between 10 1/2" and 11" is most comfortable.
Sorry, this is wordy because of the complex nature of stairs.
To determine the 2 dimensional space required, take the overall measurement from floor to floor as said above, for a straight run of stairs. Divide by 7" to get the number of risers you will need. It most likely won't be even, so adjust the number of risers up or down to get your number. Say the floor to floor number is 104 1/2". Divided by 7 you would get approx. 14 15/16 risers. Hard to do so 104 1/2"/ by 15 risers = approx a rise of 16 15/16". 14 rises would be approx. 7 7/16"
For 14 risers you would need 13 treads at 10" for approx.10'-10" in length or the 10 1/2" tread preferred for approx. 11' 4 1/2". Now remember overall stair length won't necessarily be the same length you have to have for the opening you will cut into the upper floor. That size may be a little shorter as long as you have the headroom mentioned above of at least 80"
Clear as mud? There are many factors to be determined to have the vision to understand how it will all come together before cutting into any floor joists.
Let me know if I can add anything...or clarify. You may need to have a professional be onsite to give you the best assessment.