Woah.. Do NOT check output voltage. That's running in the 30k range, and while it's VERY small amperage, it's enough to cause serious harm.
It can be the transformer, but it's likely it's the tube.
An important bit of info here needed is this -
If your tube Argon or Neon? (If it's cred when lit, clear when off, it's Neon. If it's white or gold or another color when off, it's Argon.)
If it's Neon, then the only light you're seeing is the red neon gas burning. If that's getting dim, then power to the tube is dropping - either the gas has been thinned out, the electrodes are failing, or the transformer's output voltage is dropping.
If it's Argon, then the above issues with Neon apply, but the issue is LIKELY be the phosphor in the tube. Argon is a white-burning gas, and the phosphor coating on the tube gives the tube it's color. After a long time of use, the phosphor can degrade and output less light.
Since the transformer is fed by the game's +12v, I would, *first*, check the voltage getting to the transformer. Check the voltage drop when the neon should be on, and make sure it's not low. (If memory recalls, those transformers accept 9-14v input. While that means the neon would work between 9v and 14v, lower voltages coming in means less voltage going out, which will change the brightness of the tube.)
If you really want a new transformer, I'll pull one of mine and give you the specs tonight - anything that matches can be used, but may not fit in the end of the ramp.