What year model is the current Mac Mini?
While the old Mac Mini maybe reliable at the moment and File Sharing would work via Ethernet or your local network for accessing the files. The old Mac Mini could go kaput at any moment and then you would need to find a way to extract the data or the drive from the old Mac Mini.
IMHO you really need to consider moving the data from the old Mac Mini to something safer now instead of waiting until the old Mac Mini dies and becomes a problem. At this point I'm not sure if you have a backup of the old Mac Mini, but I would recommend that you get at least 1 external drive (2 would be better). Move or Clone the data on the old Mac Mini to an external drive and then simply plug the external drive into your new Mac Mini when you need data from it.