I would assume that if the MLB All-Star game was rained out, they would play the next day. But what if it rained that day too? Would they postpone the game or cancel it? And if they canceled it, how would they decide which team gets the home field advantage in the World Series?
Not sure they’d just cancel it. My WAG would be that they’d cancel the next day’s games and have it then, then make up those games at the end of the season.
Should said monsoon go any longer than that maybe “postponed indefinitely.” In such a case I don’t think there’d be any way MLB comes out without criticism. Maybe give home field to the league that didn’t have it last year and have the All-Star game at some warmer ballpark during the winter.
Give home field advantage to whichever league won the Home Run Derby? 
My guess is that they’d decide it with a coin flip. I don’t think (and certainly hope) they would cancel regular season games just to get in an exhibition game. Postponing regular season games this time of year should not be done unless absolutely necessary. You’re getting to the time of year when teams don’t have as many dates to make up games. I don’t think there is anyone, outside of the owners who likes the day-night doubleheader.
The 1969 All Star game in Washington DC was originally rained out, and was made up the following day. This was well before the “this time it counts” nonsense, but the game was taken seriously enough anyway.
That year’s AL starting pitcher, Detroit’s Denny McLain, got into a lot of trouble because he’d scheduled a dental appointment for the open day after the all star game, thinking that, naturally enough, that the open day after the ASG would be a good day to have the appointment. When the game got pushed back a day due to the rain out, McLain went to his dentist in Detroit anyway. McLain was a pilot and figured that he could go to the appointment and then fly to Washington in plenty of time to make the game. Unfortunately for McLain, bad weather extended his flight time and he didn’t make it to the stadium until the 4th inning.