If you blast it, a bunch is likely to get sucked into Earth’s gravity well and land on us. Many smaller catastrophes is not a desired outcome when the probability of a big catastrophe is in the hundredths of a percent.
I will say it again, I think that in a few decades, we will be in a position of knowing everything that could possibly pose a danger to Earth, as well as being able to move or blast them out of the way, as may be appropriate in any given scenario. We will have many bases in near Earth orbit, probably several in geosynchronous orbit, some on the Moon and something on Mars. Between all of those, we will be able to see lots more and reach lots more than we currently are able to do. In my mind, it is pretty likely that AI will take care of this for us without us having to do much, if anything, until the problem has been solved.
If you blast the ‘roid (sounds like a name for a really vile punk rock band), the pieces will virtually all be on a new / different vector. Assuming the asteroid was on course to hit the Earth or pass very close by, so long as you hit it correctly at a respectable distance*, the redirected fragments will miss the Earth.
*Space is really BIG. Out by the asteroid belt should be fine. The orbit of Jupiter might be better, as Jupiter would tend to harvest the fragments over time. Plus, should the mission fail, there might be time for a 2nd attempt.