Do I need to cook homemade brown beer mustard?

It is probably fine without extra cooking.

The recipes for beer mustard suggest that beer would likely be a third or less of the volume of the finished mustard (the major ingredients I saw being mustard seeds, vinegar, and any other flavorings). Beer tends to be quite low in alcohol, as a rule, averaging about 5% (I've seen 3%-8%, and a few outliers with higher %s, but 5% is fairly common). So your mustard is going to be, roughly 1.66% alcohol by volume - for comparison's sake, yeast brewed sodas are usually under 1% alcohol. That is probably kid-safe, especially since mustard is usually used by the spoonful, not cupful, the alcohol intake will be likely be drops. Now, if you pick a beer that's, say, 15% (among the highest I recall seeing), then your mustard might be as high as 5%, and if your kids take a cupful or more in a sitting, they will likely get buzzed - so some common sense should still apply.

On the other hand, it tends to take more time for alcohol to cook out than people think - like an hour or two. Alcohol does evaporate faster than water, but not enough faster that they aren't being lost at the same time, so the going is pretty slow. So a couple hours simmering will be needed to be pretty sure you got most of the alcohol out (and you will have to replace the water, or use more beer and deal with stronger flavor, something like that). And if you're absolutely militant about not having any alcohol in your recipe, if for example you're dealing with allergies, you still won't be sure of getting every drop of alcohol out of the mustard.

If you want no alcohol at all, with no wiggle-room, you could probably find a non-alcoholic mustard recipe, or possibly experiment with non-alcoholic beer in your mustard. But, if you're only worried about excess consumption - your beer mustard is likely safe even for kids, since the amount of alcohol consumed per serving is very low.

Checking out recipes for brown beer mustard online, it looks like there's a pretty even split between cooked and uncooked. Be advised that cooking will not remove all the alcohol - this question Cooking away alcohol and answers may help you should you decide to cook the mustard.

I can not answer as to how much beer would be safe for a child to consume (that would fall under health, which this site does not discuss). But you can look at how much beer is in the final product and go from there.

Example:

If beer makes up half the volume of the finished product and you have a 1 Tbsp serving, you would be consuming 1/2 Tbsp of beer (1/4 ounce), a 1/4 cup serving would have 2 Tbsp of beer (1 ounce).

The formula to determine how much in any given serving is:

Serving size X % beer in finished product = how much beer per serving.