Yes-a fork with LESS TRAVEL- say 63mm vs 100mm- will drop the standover height a bit-maybe 15mm or so 63 to 100mm
What is the travel on that fork?
If you are "handy"-there probably is a way to decrease travel-
You could even go to a rigid fork-maybe $50 used on ebay-it could drop standover as much as 30mm -depending.
Usually-drop travel by 10mm-gets you about 3-4mm lower standover
But change the tires-leave the fork alone-
You can also got to shorter tires-frequently that is simpler- 5mm less sidewall will get you 10mm lower seat height.
I would change tires-you can always use spare tires-spare forks-not as much.
So get lower tires-forget the fork change-
Those tires-pretty tall-45 vs 52-will drop you 1/2 inch-25.2mm per inch-but tires are actually "less wide high" than claimed.