Your carbs look like they are off a FJ 1100 ? There should be 2 points at which the gas will not flow when the engine is shut off. At the petcock and then the needle and seats. Looks like both have failed if it leaked while shut down in storage. While you are servicing the fuel system you might take a look at the fuel shut off at the petcock also. Depending on the year of FJ there is at least 3 different main fuel shut off systems.
Good catch on finding the problem. What year is your bike? There is a safety mod to one of the fuel shut off systems that might apply.
Here is the information on the mod for the early FJ (84-86) main fuel shut off
http://www.fjowners.com/index.php?topic=3265.0 Randy (or Robert) at RPM is about the best source of carb parts that I know of.
http://www.rpmracingca.com/products.asp?cat=24If the carbs have been sitting for 5 or 6 years they probably need a good cleaning for sure. Pay a lot attention the pilot jets. They are a bugger. There orifice is about the size of a pin head and plug easy. There is a lot of good info in the files section on the carbs
http://www.fjowners.com/index.php?board=21.0 George