Old technologies support, is it worth it?

I just realized that I will be interacting with old and apparently unsupported technologies in my new job. I will be helping to maintain a classic ASP web site.
The code standards were really hard to follow back in those days, and the developers had a lot to think about when they were building this kind of tools, but, is it worth it?

I guess depends on your point of view, from one side a really specialized application can take a lot of time to be fully documented and then developed in a new technology and then at the end decommissioned but on the other side the amount of time required just to make a quick enhancement makes me think this scenario.

I really don’t like the idea of supporting a tool that is built in a technology that because of a lot of good reasons is not longer been fully supported by the vendor, but what I can say, I’m not the one taking those kind of decisions, yet. I will keep point of view trying to suggest we move to a new technology by friendly and repeatedly share with the customer (if possible) the idea of migrating to some better technology than the one currently in place.

What do you think? do you think it is worth it? what would you do?