From my experience there is no evidence that the equation big vendor = stable company = stable platform is valid. In Europe there are lots of bridges that are still on roman fundaments. I would certainly call that a stable platform.
In IT no company has more than 20-30 years in open systems. New technologies have always one shortcoming from a risk perspective, and that is that they are new.
I found that new technologies from big vendors tend to be as (un)reliable as from any other vendor and my gut feeling is they often are worse. The smaller vendor seems to have a plus insofar as they focus on one product or technology only. Their commitment and expertise often is superior to that of a big company, where a similar product is one out of 3000 others in the pricelist. Smaller companies tend to give very good service as they have to survive and can do so only by building trust through service and customer satisfaction. Your power as a customer is much bigger if you go with a small vendor than if you go with a "big player", unless you are a big player as well. In short I often see more pluses on the small vendor side than on the "established" firm side. But at the end of the day these are only hints and can not be taken as a recipe. If you feel good about the technology and the expertise of the vendor I can not see any benefits by going with a "stable" -- as you put it – company.
Hope that helps a bit.
This was first published in June 2004