I don't personally enjoy their music, but it doesn't take a lot for me to figure out what their appeal is or was anyway...like rap's first big white act that made fun party music, makes sense in context...what I don't get so much (but have a pretty good reason as to why anyway) is how they maintained all the way through the 80's and well into the 90's with an outdated style...while their contemporaries like Run DMC and the like had long since been thrown in the bushes....they became something of proto-hipster critical darlings with an entire new generation of fans...
They embraced their "whiteness" if that makes sense. They did more of the punk-rock stuff "Sabotage", and then moved into the electronic world with Hello Nasty and To The 5 Boroughs. The late 90s seems like the time a lot of people started listening to electronic music and labels were taking it seriously. They were still rapping, but at times it almost didnt feel like hip-hop.
They also had a full artistic image (kinda like Kanye) where their videos, album covers etc all had a distinct aesthetic to them. They branded themselves very well. Hipster journalists kinda love that stuff
Most of the older groups stuck to the script, where as the Beasties knew they couldnt cut it in the hip-hop world anymore. I think they also originated as a punk band