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.