Has anybody ever replaced an entire crossover assembly (circuit board) in a Alesis Monitor One? Specifically, does the woofer need to come out, and if so, how?

What you are describing is a total component failure of something in the crossover board. You cannot really check a crossover without an oscilloscope and a frequency generator. In a lot of cheap speaker systems, there is no crossover board attached to the woofer. In your case it appears that a low pass filter is present on the woofer. http://circuitdiagram.net/loudspeaker-system-crossover-network.html

I assume you mean the Pr-12RX passive? That seems wrong to me, too. Put an ohm meter across the terminals of the horn driver and the woofer seperatly to see which, if either, is blown. If they're OK, look for a bad solder joint on the crossover.

Unfortunately Behringer equipment is hard to get parts for generically. You may be able to order the whole amp from behringer but I doubt it. I have had lots of problems repairing them for that reason and I will no longer use any of their stuff. The IC's could be repaired but can't find any record or crossover to any of their output transistors. It appears they are custom made for Behringer. Sorry, I can't help any more than that.

The Problem is a dead battery which is soldered to the circuit board inside the unit. It's a Panasonic BR-2325-1HC, but really you can replace it with any similar 3V battery.
Here's how I fixed mine.
http://andrewapeterson.com/2013/08/fix-alesis-q2-first-made-strange-sounds-then-bad-program-issue-quadraverb-2-battery/

Crossovers RARELY blow... most likely the tweeter speakers have blown... not fun to find replacements.

Unscrew the woofer, gently pull it out, follow the wires to the crossover, replace the fuse, reverse the process.

This speaker is not an active one. This means that it hasn't integrated amplifier, so you need an external one. The cable from the amplifier goes to the speaker. Remove the woofer in order to get access at the inner parts and check these points:

• In case that there is a crossover present there must be a burned out coil or capacitance
• In case the cable from the female jacks at the back panel goes direct to the woofer, there must be the woofer destroyed or the cable disconnected. Check the cables, it's less expensive than to replace the woofer.

Post me back if there is a crossover in use or it is direct connected and any other points of interest you came up, in order to guide you further.
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Stelios
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The crossover in these is very simple and if it is burned, likely your speakers, maybe both, the woofer and tweeter have been damaged and need to be replaced. Test the speakers before changing the crossover. You MAY NOT notice the damage until you start driving the speakers again. The voice coil of the woofer is a common thing to fry, and while it may not open up, often it becomes unbonded to the coil form and rubs the magnet pole pieces. The tweeter/horn often wipes out the diaphram or blows the element open. Some of these are piezo horns and they can sometimes survive. These are rated at 100 Watts and if your amp can deliver over 50 watts you have to be careful to not put too much power into these. I have several of the Kustom 10 inch ones and several of the 15 inch ones... Also I have two 15 double stacks and they all seem to use the same crossover. Believe me, the crossover is far from what I would call "special" like the advertisements say. As an electrical engineer I was very disappointed. Anyway, test your speakers as they may be the worst of the problem.

The question is WHY would you replace the crossover network?

They seldom go bad... Usually the speakers aree blown before the crossover fails.

I would recommend using a separate active crossover such as the Peavey 23XO and a separate amp for your subs.