The SNES is a rather robust machine; however, there are still fragile components inside. There are a few things to check for:
1) Check to see if you are actually drawing power from the AC adapter. Cheap, 3rd-party adapters are not noted for their longevity, and the original Nintendo ones are quite ancient now as well.
2) Check the power input jack on the SNES. There should be a pair of metal teeth inside a plastic circular enclosure. If that enclosure is broken, it may not be making good contact with your AC adapter.
3) The fuse that is in the SNES is notorious for breaking. Every time I get a new SNES in, I'll replace the picofuse with a compatible (125V, 1.5A) resettable polyfuse. Basically, if there is a power surge that blows the fuse, all I'll need to do is turn the SNES off and then on again to reset the fuse, instead of having to replace the fuse.