I was given a Dell monitor (E178FPb) for free, because it quit working. I would like to try to repair it if at all possible but I'm not sure what to look for. When the monitor is plugged in, the LED for the power button blinks rapidly, over and over. Any suggestions? Can it be fixed?

It CAN be fixed.....problem is is it worth fixing. And many times its not. Not after you buy schematics, assuming you own or can borrow the needed test gear, and know what to do with all of them....and find out what is actually defective sometimes its cheaper to buy a new one than fix an old one.. Sometimes you get lucky and its a cheap part that fails.

Okay. Assuming I want to at least TRY to fix it, is there anything specific that I should look for? I saw two swollen capacitors on the mainboard. Should I start by replacing those?

Swollen capacitors might contribute to it but I highly doubt they are a problem by themselves. But without knowing exactly what function they serve in the circuit its hard do say for sure....they are cheap to replace however. Match the capacitance and voltage ratings. As well as the all important type. They aren't all the same nor or they generally interchangible between types.