Circuit board fuse blows on AC switch
Without knowing much about your system, I'd recommend starting by checking the contactor in the condensing unit  (outdoor unit).
Before you begin
Before you open up the condensing unit, make sure to turn the power off at the breaker, pull/turn off the serviceman switch, and verify power is off.
Once you've opened the unit, you'll want to properly discharge the capacitor(s).  You don't want them to accidentally discharge through you.
Check the contactor
Using an insulated screwdriver, or pair of pliers, push the contactor plate in. You'll want to make sure the plate moves freely, and is not sticking or getting hung up.
If you're really careful, and comfortable doing so, you can run the system with the condensing unit open. Then you'll be able to see if the contactor pulls in.
Here's a good YouTube video that shows why the contactor not pulling in is a problem.
Next, set your multimeter to test resistance.  Carefully remove the leads from the coil terminals, then test the resistance of the coil by touching one probe on each of the coil contacts (where you removed the wires). Make sure the resistance matches the manufacturer's specifications.
If the resistance is really low, the coil is likely shorted out.  If the resistance is infinity, the coil is likely broken.  In both cases, the contactor will need to be replaced.

Isolate the problem
To try and determine where the problem is,  you could disconnect the control wires feeding the outside unit. Then switch the unit to COOL, and lower the set point until the thermostat calls for COOL.  If the fuse blows with the outdoor unit disconnected, then the problem is likely on the control board.
Unfortunately the control board doesn't appear to be serviceable, so you'll have to replace the entire board if that's where the problem is.