Reiterating what others have said, neither of the two Disney golf courses is necessarily closer to the Magic Kingdom. Additionally, you did not indicate that you would be taking a car, so I'm assuming you're asking about Disney transportation. From your question, it looks like what you are perhaps looking for is a way to get quickly from the Magic Kingdom to a mini-golf course. In this case, your options are:
Fantasia Gardens (my personal favorite): Take a bus from MK to the Swan Resort. Get off the bus and walk across the street to where the mini golf place is. There should be signs and a sidewalk/crosswalk.
Winter Summerland: You will need to first go to a resort. The Contemporary is the closest resort to the MK (just one monorail stop away). Alternatively, you could return to your own resort. From there, you need to hop a bus to Blizzard Beach. Note that you will not be able to get a bus to Blizzard Beach after Blizzard Beach has closed (even though Winter Summerland may still be open). Winter Summerland is just outside (and to the right) of the Blizzard Beach entrance.
Another thing to consider is, "Once I'm done mini-golfing how do I get back to my resort/a park?" For Fantasia Gardens you would need to return to the Swan and catch a bus to a park (or the boat, if you want to go to Epcot or MGM). From a park, you can get to your resort. For Winter Summerland, you should be able to pick up a bus to your resort from the Blizzard Beach stop, however, I do not believe there are any direct park buses from there (except to Animal Kingdom). As such, to get to a park, you'd have to go to a resort and take a bus from there.