The Newel Post (as it's called) ... It may not be a very 'sightly' fix if you don't remove it to see what it is that's suppose to be holding the post in place.
Usually, unless you have direct access to the floor, either from below or from above, your only options is to replace the Newel Post. It's quite possible that the 'pin' that the post sits on has 'rounded-out' the hole in the post. Maybe the bolt that holds the post to the joist is coming loose.
I would suggest you start by looking up from the finished basement. You only need a small hole (assuming it's drywall) to see what's up there. A small hole can be easily repaired. The hole needs to be big enough to shine a light into AND look w/your eye, at the same time. (Post back if you need info on how to repair a hole in drywall.) IF you do this, use a Utility Knife and be CAREFUL when you cut! THere could be electrical lines and plumbing lines up there that you can't see. If the post can be repaired from below, you'll need a larger hole. If not, then based on what you CAN see w/regards to the post (from below), either repair or replace the post.
Do your best in trying to 'identify' the location (in the basement) where you need to cut the hole. The difference in measurements of 1-3/4" could mean you're on the WRONG side of a joist, and you'll need to cut another hole. (And on and on and on ...) Sometimes, if you pick the right place, you can drill a hole in the floor (from above) that's near the post (or at the post) that's big enough to put a coat hanger in. Then, you locate that 'spot' in the basement, and measure over to the post accordingly. (I've done this w/great accuracy as long as the 'line' from my test-hole to the post is a straight one ...)