Judy,
I don't know whether it's the closest to Universal (I am directionally challenged), but we have played once at Disney's Fantasia Gardens when we stayed at WDW Dolphin last Christmas. It was very pleasant. Went there during the day and it was not too crowded. Theming is fun as well!! I am not a connoisseur of mini golf, but we had a blast! I suppose you would need a taxi for that though...