FYI - The voltage divider CAN'T blow the fuse because the current through the voltage divider doesn't go through the fuse... And ~1mA is what you'd expect (12V/13.3K) so if the values are correct and it's wired correctly it won't draw enough current to blow the fuse. And, I assume you are also measuring about 3.3V.
All of the grounds should be connected together. But, missing grounds would not cause a blown fuse. Of course a wrong ground connection... A short to ground... can blow a fuse or cause other problems.
Too much current would indicate a bad or mis-wired voltage regulator module or a mis-wire to the Arduino, or the Arduino is simply bad (and shorted out).
Since you apparently have a multimeter, disconnect the Arduino and if you have time before the fuse blows, check the voltage into and out-of the LM1117 voltage regulator. If it's OK, check the current into the voltage regulator without the Arduino connected. If you are getting excess current without the Arduino, the voltage regulator is dead (shorted) or mis-wired.
If you are getting excess current through the Arduino it's most-likely dead, although it might be mis-wired in a way that draws excess current without getting destroyed.
Usually when a fuse blows, something else has already blown. There's an old joke, "The transistor blew to protect the fuse". the purpose of a fuse is usually to prevent a fire, or to prevent further damage after something has gone wrong. The fuse in your circuit should protect the power supply but it may not protect the Arduino.
Your schematic doesn't show any connected Arduino outputs. A microcontroller can be useful with no inputs, but it's kind-of useless without outputs. I assume there are outputs (unless maybe just the USB serial monitor or watching a built-in LED?) and maybe the problem is with something you didn't show us.
You didn't say which Arduino you're using, but the standard Uno has a built-in voltage regulator so you can power it directly off of 12V as long as you are ONLY powering the Arduino (and maybe a few LEDs, etc.) through that built-in regulator.
Logic-high is "nominally" +5V. 3.3V will work but it's "cutting it close" for a logic-high. Anything greater than 3V (0.6Vcc) is read as logic high and anything less than 1.5V (0.3Vcc) is read as low. Between 1.5 and 3V is undefined and it may read high or low.
The "absolute maximum" to an input is 5.5V (Vcc+0.5V), but there are (low current) over-voltage protection diodes on the I/O pins so with the 10K voltage divider resistance limiting current, you can actually go over a "calculated" 5.5V, and the protection diodes will safely limit the voltage. So IF there is current limiting, you don't have to be too paranoid about accidently putting more than 5V on an input pin.