So by old I'm talking maybe ten years! I was cleaning and found quite a bit of LME gifted to me with all my original homebrew equipment. I was mid-swing into the trash can when I wondered if maybe I could use this old stuff for yeast starters? A good boil should clean up any nasties in there right? It's been sealed in a plastic screw-top jar that was taped and some in a can. It sure would be a penny saver if I could use it, what do you think?
Sure, dilute it, boil it to get about 1.040 SG and use it in the starter. All of the sugar content is still there. It may have staled and developed soapy compounds but the yeast don't care about those - they just want the sugar!
After making the starter, put it in the fridge to flocculate the yeast and pour off the liquid so none of it makes it into the beer.
If you have access to canning equipment, can the rest so that it's ready for starters for subsequent brews.
Doing some searching around, LME can go bad, but I don't know how much it would effect your beer if it's just a yeast starter. You apparently get what people have dubbed "Extract Twang" if the LME is shelved for too long, as well as the color of the extract darkens, but I'd imagine a yeast starter is insignificant enough in quantity that you probably don't have to worry much.
If I had to choose between the two, I'd say the canned is probably your best bet, as it sounds as though the plastic jar may be permeable to oxygen, therefor making it the least preserved over time between the two. I'd say use up the canned LME in your starters before using the plastic container, and if you don't notice any off smell/flavors with the canned starter/beer, you probably won't notice any issues with the plastic jar. That is of course, assuming the plastic jar isn't being re-opened and rotting from previous exposure to the elements.
Just bear in mind, I assume you won't use all of the extract in one starter, as soon as you open it, you're already at a disadvantage, so make sure the remaining extract is refrigerated, and used before very long after opening the old can/jar. While I'm guessing at this, I wouldn't hold onto it for more than 2 months after opening it, and even that seems to be a bit long.